A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Let $\mathcal{O}_n$ be the set of all discrete t-operators on $L_n$. A formula for $|\mathcal{O}_n|$ can be obtained in a manner similar to the formula derived for some classes of discrete uninorms, explained in Remark 3.5, also depending on the number of discrete t-norms and discrete t-conorms. This formula is given by

$$|\mathcal{O}_n| = |S_n| + |\mathcal{T}_n| + \sum_{k=1}^{n-1} |S_k| \cdot |\mathcal{T}_{n-k}| = 2 \cdot |\mathcal{T}_n| + \sum_{e=1}^{n-1} |\mathcal{T}_e| \cdot |\mathcal{T}_{n-e}|, \quad (50)$$

which interestingly coincides with the formula for $|\mathcal{U}_n^{\text{MIN}}|$ and $|\mathcal{U}_n^{\text{MAX}}|$. But further consequences can be drawn from Theorem 3.8. Mainly, that a discrete t-operator is smooth if, and only if, the underlying discrete t-norm and discrete t-conorm are smooth, a result also reached by Mas et al. in [7] with discrete t-operators. If we substitute $|\mathcal{T}_n|$ in Equation (50) with $|\mathcal{T}_n^{\text{SMT}}|$, we obtain a formula for the number of smooth discrete t-operators, which is equivalent to the number of discrete t-operators. And finally, if we look for the number of discrete t-operators that are idempotent, given that there is only one idempotent discrete t-norm and t-conorm, we can obtain a formula for this class of discrete t-operators.

**Proposition 3.21.** ([7, Corollary 3.8]) For $n \ge 2$, the following statements hold:

(i) If $\mathcal{O}_n^{\text{SMT}}$ denotes the set of smooth discrete t-operators, then $|\mathcal{O}_n^{\text{SMT}}| = 2^n + (n-1) \cdot 2^{n-2}$.

(ii) If $\mathcal{O}_n^{\text{IDE}}$ denotes the set of idempotent discrete t-operators, then $|\mathcal{O}_n^{\text{IDE}}| = n + 1$.

We finish this section with discrete nullnorms, introduced and characterized by Riera and Torrens in [59].

**Definition 3.17.** ([59, Definition 2.4]) A discrete nullnorm is an associative and commutative discrete aggregation function $H: L_n^2 \to L_n$ which has an absorbing element $k \in L_n$ satisfying $H(0, x) = x$, for all $x \le k$, and $H(n, x) = x$, for all $x \ge k$.

In a similar way to what happens on [0, 1], discrete nullnorms and discrete t-operators have different definitions (albeit with shared properties), Riera and Torrens characterized in [59, Proposition 2.7] discrete nullnorms with the same result as the one stated in Theorem 3.8; that is, an operator $H$ is a discrete nullnorm if, and only if, it satisfies Equation (49). Given that these two classes of operators are essentially the same, it is evident that the enumerations discussed previously are also applicable to discrete nullnorms. Specifically, the number of discrete nullnorms, $|H_n|$, is governed by the formula of Equation (50). Moreover, $|H_n^{\text{SMT}}|$ and $|H_n^{\text{IDE}}|$, representing the number of smooth and idempotent discrete nullnorms, respectively, are determined in Proposition 3.21. Table 7 provides examples of the enumerations discussed in this section for $n \le 13$.

# 4. Enumeration of discrete implications and some subclasses

Discrete implications extend the concept of implication from Boolean logic, mirroring how discrete negations, discrete t-norms and discrete t-conorms broaden the notions of negation, conjunction and disjunction in Boolean logic, respectively. The two surveys [60, 61] compile a large number of properties of fuzzy implication functions<fnref n="2" />, while in the two surveys [62, 63] we find sections dedicated to discrete implications. Initially, the concept of discrete implication was proposed by Nachtegael and Kerre [6] under the name of *implicator*, and was studied in detail by Mas et al. in [64] establishing the usual nomenclature.

**Definition 4.1.** ([64, Definition 3]) A discrete implication is a binary operation $I: L_n^2 \to L_n$ such that it is decreasing in the first argument, increasing in the second argument, $I(0, 0) = I(n, n) = n$ and $I(n, 0) = 0$.

The cardinality of these operators already attracted the attention of Nachtegael and Kerre in their seminal paper [6]. Computationally, they determined the number of discrete implications on $L_n$ for $n \le 5$, and it was not until years later that Munar et al. in [19] determined a general formula for the enumeration of discrete implications. Their procedure was based on establishing a bijection between discrete implications and discrete conjunctions.

<footnote n="2">²When the domain is [0, 1], the implications defined on it are generally referred to as *fuzzy implications*.</footnote>
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 7**</caption><thead><tr><th>n</th><th>Ο<sub>n</sub>, H<sub>n</sub></th><th>Ο<sub>n</sub><sup>SMT</sup>, H<sub>n</sub><sup>SMT</sup></th><th>Ο<sub>n</sub><sup>IDE</sup>, H<sub>n</sub><sup>IDE</sup></th></tr></thead><tbody><tr><td>1</td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td></tr><tr><td>2</td><td>5 = 5 · 10<sup>0</sup></td><td>5 = 5 · 10<sup>0</sup></td><td>3 = 3 · 10<sup>0</sup></td></tr><tr><td>3</td><td>16 = 1.6 · 10<sup>1</sup></td><td>12 = 1.2 · 10<sup>1</sup></td><td>4 = 4 · 10<sup>0</sup></td></tr><tr><td>4</td><td>60 = 6 · 10<sup>1</sup></td><td>28 = 2.8 · 10<sup>1</sup></td><td>5 = 5 · 10<sup>0</sup></td></tr><tr><td>5</td><td>256 = 2.56 · 10<sup>2</sup></td><td>64 = 6.4 · 10<sup>1</sup></td><td>6 = 6 · 10<sup>0</sup></td></tr><tr><td>6</td><td>1,214 = 1.214 · 10<sup>3</sup></td><td>144 = 1.44 · 10<sup>2</sup></td><td>7 = 7 · 10<sup>0</sup></td></tr><tr><td>7</td><td>6,314 = 6.314 · 10<sup>3</sup></td><td>320 = 3.2 · 10<sup>2</sup></td><td>8 = 8 · 10<sup>0</sup></td></tr><tr><td>8</td><td>35,738 = 3.5738 · 10<sup>4</sup></td><td>704 = 7.04 · 10<sup>2</sup></td><td>9 = 9 · 10<sup>0</sup></td></tr><tr><td>9</td><td>219,476 ≈ 2.1948 · 10<sup>5</sup></td><td>1,536 = 1.536 · 10<sup>3</sup></td><td>10 = 1 · 10<sup>1</sup></td></tr><tr><td>10</td><td>1,466,224 ≈ 1.4662 · 10<sup>6</sup></td><td>3,328 = 3.328 · 10<sup>3</sup></td><td>11 = 1.1 · 10<sup>1</sup></td></tr><tr><td>11</td><td>10,773,776 ≈ 1.0774 · 10<sup>7</sup></td><td>7,168 = 7.168 · 10<sup>3</sup></td><td>12 = 1.2 · 10<sup>1</sup></td></tr><tr><td>12</td><td>89,287,985 ≈ 8.9288 · 10<sup>7</sup></td><td>15,360 = 1.536 · 10<sup>4</sup></td><td>13 = 1.3 · 10<sup>1</sup></td></tr><tr><td>13</td><td>874,252,030 ≈ 8.7425 · 10<sup>8</sup></td><td>32,768 = 3.2768 · 10<sup>4</sup></td><td>14 = 1.4 · 10<sup>1</sup></td></tr></tbody></table>

Enumeration of some classes of discrete t-operators and nullnorms, for $n \le 13$. From left to right: discrete t-operators and nullnorms in general, obtained with Equation (50), which depends on the number of discrete t-norms in general, which is unknown; smooth discrete t-operators and nullnorms, given by Proposition 3.21; and finally idempotent discrete t-operators and nullnorms, given by Proposition 3.21.

**Proposition 4.1.** ([19, Proposition 4.1]) Let $n \ge 1$, and let us denote by $C_n$ and $I_n$ the sets of discrete conjunctions and discrete implications, respectively, defined on $L_n$. Let $\varphi: C_n \to I_n$ be defined point-wise by

$$ \varphi(C)(x, y) = n - C(x, n - y), $$

for all $x, y \in L_n$, where $C \in C_n$. Then, $\varphi(C)$ is a discrete implication, and the transformation $\varphi$ is a bijection.

From this result, we have the following consequences. Firstly, that the number of discrete implications defined on $L_n$, $|I_n|$, coincides with $|C_n|$, which is given in Proposition 3.5 (see [19, Corollary 4.1]). Secondly, if we denote by $C_n^{\text{SMT}}$ and $I_n^{\text{SMT}}$ the sets of smooth discrete conjunctions and smooth discrete implications, respectively, the same formula of $\varphi$ given in Proposition 4.1 but restricted to these sets of smooth operators also remains a bijection, so that $I_n^{\text{SMT}}$ is given by Proposition 3.9 (see [20, Corollary 3.2.2]).

The definition of discrete implication is quite generic, and as such, it does not allow to derive certain properties which are useful in different situations. Consequently, the literature has proposed several additional properties for discrete implications, and also their cardinality has been investigated. The main ones are listed below.

* The contrapositive symmetry with respect to a discrete negation $N$,

$$ I(x, y) = I(N(y), N(x)), \quad x, y \in L_n. \qquad (\text{CP}(N)) $$

* The left neutrality principle,

$$ I(n, y) = y, \quad y \in L_n. \qquad (\text{NP}) $$

* The consequent boundary,

$$ I(x, y) \ge y, \quad x, y \in L_n. \qquad (\text{CB}) $$

* The ordering principle,

$$ I(x, y) = n \Leftrightarrow x \le y, \quad x, y \in L_n. \qquad (\text{OP}) $$

* The identity principle,

$$ I(x, x) = n, \quad x \in L_n. \qquad (\text{IP}) $$

* The exchange principle,

$$ I(x, I(y, z)) = I(y, I(x, z)), \quad x, y, z \in L_n. \qquad (\text{EP}) $$
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 7**</caption><thead><tr><th>n</th><th>Ο<sub>n</sub>, H<sub>n</sub></th><th>Ο<sup>SMT</sup><sub>n</sub>, H<sup>SMT</sup><sub>n</sub></th><th>Ο<sup>IDE</sup><sub>n</sub>, H<sup>IDE</sup><sub>n</sub></th></tr></thead><tbody><tr><td>1</td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td></tr><tr><td>2</td><td>5 = 5 · 10<sup>0</sup></td><td>5 = 5 · 10<sup>0</sup></td><td>3 = 3 · 10<sup>0</sup></td></tr><tr><td>3</td><td>16 = 1.6 · 10<sup>1</sup></td><td>12 = 1.2 · 10<sup>1</sup></td><td>4 = 4 · 10<sup>0</sup></td></tr><tr><td>4</td><td>60 = 6 · 10<sup>1</sup></td><td>28 = 2.8 · 10<sup>1</sup></td><td>5 = 5 · 10<sup>0</sup></td></tr><tr><td>5</td><td>256 = 2.56 · 10<sup>2</sup></td><td>64 = 6.4 · 10<sup>1</sup></td><td>6 = 6 · 10<sup>0</sup></td></tr><tr><td>6</td><td>1,214 = 1.214 · 10<sup>3</sup></td><td>144 = 1.44 · 10<sup>2</sup></td><td>7 = 7 · 10<sup>0</sup></td></tr><tr><td>7</td><td>6,314 = 6.314 · 10<sup>3</sup></td><td>320 = 3.2 · 10<sup>2</sup></td><td>8 = 8 · 10<sup>0</sup></td></tr><tr><td>8</td><td>35,738 = 3.5738 · 10<sup>4</sup></td><td>704 = 7.04 · 10<sup>2</sup></td><td>9 = 9 · 10<sup>0</sup></td></tr><tr><td>9</td><td>219,476 ≈ 2.1948 · 10<sup>5</sup></td><td>1,536 = 1.536 · 10<sup>3</sup></td><td>10 = 1 · 10<sup>1</sup></td></tr><tr><td>10</td><td>1,466,224 ≈ 1.4662 · 10<sup>6</sup></td><td>3,328 = 3.328 · 10<sup>3</sup></td><td>11 = 1.1 · 10<sup>1</sup></td></tr><tr><td>11</td><td>10,773,776 ≈ 1.0774 · 10<sup>7</sup></td><td>7,168 = 7.168 · 10<sup>3</sup></td><td>12 = 1.2 · 10<sup>1</sup></td></tr><tr><td>12</td><td>89,287,985 ≈ 8.9288 · 10<sup>7</sup></td><td>15,360 = 1.536 · 10<sup>4</sup></td><td>13 = 1.3 · 10<sup>1</sup></td></tr><tr><td>13</td><td>874,252,030 ≈ 8.7425 · 10<sup>8</sup></td><td>32,768 = 3.2768 · 10<sup>4</sup></td><td>14 = 1.4 · 10<sup>1</sup></td></tr></tbody></table>

Enumeration of some classes of discrete t-operators and nullnorms, for $n \le 13$. From left to right: discrete t-operators and nullnorms in general, obtained with Equation (50), which depends on the number of discrete t-norms in general, which is unknown; smooth discrete t-operators and nullnorms, given by Proposition 3.21; and finally idempotent discrete t-operators and nullnorms, given by Proposition 3.21.

**Proposition 4.1.** ([19, Proposition 4.1]) Let $n \ge 1$, and let us denote by $C_n$ and $I_n$ the sets of discrete conjunctions and discrete implications, respectively, defined on $L_n$. Let $\varphi : C_n \to I_n$ be defined point-wise by

$$ \varphi(C)(x, y) = n - C(x, n - y), $$

for all $x, y \in L_n$, where $C \in C_n$. Then, $\varphi(C)$ is a discrete implication, and the transformation $\varphi$ is a bijection.

From this result, we have the following consequences. Firstly, that the number of discrete implications defined on $L_n$, $|I_n|$, coincides with $|C_n|$, which is given in Proposition 3.5 (see [19, Corollary 4.1]). Secondly, if we denote by $C_n^{SMT}$ and $I_n^{SMT}$ the sets of smooth discrete conjunctions and smooth discrete implications, respectively, the same formula of $\varphi$ given in Proposition 4.1 but restricted to these sets of smooth operators also remains a bijection, so that $I_n^{SMT}$ is given by Proposition 3.9 (see [20, Corollary 3.2.2]).

The definition of discrete implication is quite generic, and as such, it does not allow to derive certain properties which are useful in different situations. Consequently, the literature has proposed several additional properties for discrete implications, and also their cardinality has been investigated. The main ones are listed below.

* The contrapositive symmetry with respect to a discrete negation $N$,

$$ I(x, y) = I(N(y), N(x)), \quad x, y \in L_n. \qquad (\text{CP}(N)) $$

* The left neutrality principle,

$$ I(n, y) = y, \quad y \in L_n. \qquad (\text{NP}) $$

* The consequent boundary,

$$ I(x, y) \ge y, \quad x, y \in L_n. \qquad (\text{CB}) $$

* The ordering principle,

$$ I(x, y) = n \Leftrightarrow x \le y, \quad x, y \in L_n. \qquad (\text{OP}) $$

* The identity principle,

$$ I(x, x) = n, \quad x \in L_n. \qquad (\text{IP}) $$

* The exchange principle,

$$ I(x, I(y, z)) = I(y, I(x, z)), \quad x, y, z \in L_n. \qquad (\text{EP}) $$
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

* The generalized *modus ponens*, with respect to a discrete t-norm $T$,

$$T(x, I(x, y)) \leq y, \text{ for all } x, y \in L_n. \qquad (\text{MP}(T))$$

* The generalized *modus tollens*, with respect to a discrete t-norm $T$ and a discrete negation $N$,

$$T(N(y), I(x, y)) \leq N(x), \text{ for all } x, y \in L_n. \qquad (\text{MT}(T, N))$$

* The law of importation, with respect to a discrete t-norm $T$,

$$I(T(x, y), z) = I(x, I(y, z)), \text{ for all } x, y, z \in L_n. \qquad (\text{LI}(T))$$

In the following sections we compile enumerations of discrete implications satisfying some of the above properties. In particular, formulas are known for $(\text{CP}(N_C))$ being $N_C$ the classical discrete negation, $(\text{NP})$, $(\text{CB})$, $(\text{OP})$ and $(\text{IP})$. In Table 8 we list the first values of these enumerations. For the remaining properties, a general enumeration remains an open problem.

## 4.1. Discrete implications satisfying $(\text{CP}(N))$

We start with the contrapositive symmetry with respect to a discrete negation $N$. In [19, Proposition 4.6] it was observed that the transformation $\varphi$ of Proposition 4.1 establishes a bijection between $C_n^{\text{COM}}$ and $I_n^{\text{CP}(N_C)}$, which denotes the set of discrete implications satisfying $(\text{CP}(N_C))$, i.e., satisfying the contrapositive symmetry with respect to the only strong discrete negation, $N_C(x) = n - x$ for all $x \in L_n$. For this reason, the enumeration of $I_n^{\text{CP}(N_C)}$ is given by Proposition 3.6. In addition to $N_C$, it is also possible to determine the number of discrete implications satisfying $(\text{CP}(N_{D_1}))$ and $(\text{CP}(N_{D_2}))$, where $N_{D_1}$ and $N_{D_2}$ are the least and the greatest discrete negations, respectively, and are given by

$$N_{D_1}(x) = \begin{cases} n, & \text{if } x = 0, \\ 0, & \text{otherwise,} \end{cases} \qquad N_{D_2}(x) = \begin{cases} n, & \text{if } x \neq n, \\ 0, & \text{otherwise,} \end{cases}$$

for all $x \in L_n$. Imposing the functional equation of $(\text{CP}(N))$ for each discrete negation and distinguishing different cases according to the position of $x, y \in L_n$, we have that there is only one discrete implication that satisfies $(\text{CP}(N_{D_1}))$, and it is given by

$$I_1(x, y) = \begin{cases} 0, & \text{if } y = 0 \text{ and } 1 \leq x \leq n, \\ n, & \text{otherwise,} \end{cases}$$

for all $x, y \in L_n$. Similarly, we find that there is only one discrete implication that satisfies $(\text{CP}(N_{D_2}))$, and it is given by

$$I_2(x, y) = \begin{cases} 0, & \text{if } x = n \text{ and } 0 \leq y \leq n - 1, \\ n, & \text{otherwise,} \end{cases}$$

for all $x, y \in L_n$. Beyond these cases, to date, there is no formula for $|I_n^{\text{CP}}|$ in general, i.e., the number of discrete implications satisfying $(\text{CP}(N))$ with respect to some discrete negation $N$, nor is there a computationally obtained enumeration of it.

## 4.2. Discrete implications satisfying $(\text{NP})$

Nachtegael and Kerre [6] conducted a computational analysis to determine the number of discrete implications satisfying $(\text{NP})$, denoted as $I_n^{\text{NP}}$, for $n \leq 6$. Subsequently, Munar et al. in [19] introduced a recursive formula to calculate $|I_n^{\text{NP}}|$ for any value of $n$. This approach also relied on plane partitions, specifically on those called $(\alpha, \beta)$-monotonic plane partitions, which were initially proposed by Krattenthaler in [65, 66]. These plane partitions can be described as those satisfying $\pi_{i,j} \geq \pi_{i,j+1} + \alpha$ and $\pi_{i,j} \geq \pi_{i+1,j} + \beta$ (following the same nomenclature as in Definition 3.4) whenever both sides are defined. Krattenthaler [65]
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

* The generalized *modus ponens*, with respect to a discrete t-norm $T$,

$$T(x, I(x, y)) \leq y, \text{ for all } x, y \in L_n. \qquad (\text{MP}(T))$$

* The generalized *modus tollens*, with respect to a discrete t-norm $T$ and a discrete negation $N$,

$$T(N(y), I(x, y)) \leq N(x), \text{ for all } x, y \in L_n. \qquad (\text{MT}(T, N))$$

* The law of importation, with respect to a discrete t-norm $T$,

$$I(T(x, y), z) = I(x, I(y, z)), \text{ for all } x, y, z \in L_n. \qquad (\text{LI}(T))$$

In the following sections we compile enumerations of discrete implications satisfying some of the above properties. In particular, formulas are known for $(\text{CP}(N_C))$ being $N_C$ the classical discrete negation, $(\text{NP})$, $(\text{CB})$, $(\text{OP})$ and $(\text{IP})$. In Table 8 we list the first values of these enumerations. For the remaining properties, a general enumeration remains an open problem.

## 4.1. Discrete implications satisfying $(\text{CP}(N))$

We start with the contrapositive symmetry with respect to a discrete negation $N$. In [19, Proposition 4.6] it was observed that the transformation $\varphi$ of Proposition 4.1 establishes a bijection between $C_n^{\text{COM}}$ and $I_n^{\text{CP}(N_C)}$, which denotes the set of discrete implications satisfying $(\text{CP}(N_C))$, i.e., satisfying the contrapositive symmetry with respect to the only strong discrete negation, $N_C(x) = n - x$ for all $x \in L_n$. For this reason, the enumeration of $I_n^{\text{CP}(N_C)}$ is given by Proposition 3.6. In addition to $N_C$, it is also possible to determine the number of discrete implications satisfying $(\text{CP}(N_{D_1}))$ and $(\text{CP}(N_{D_2}))$, where $N_{D_1}$ and $N_{D_2}$ are the least and the greatest discrete negations, respectively, and are given by

$$N_{D_1}(x) = \begin{cases} n, & \text{if } x = 0, \\ 0, & \text{otherwise,} \end{cases} \qquad N_{D_2}(x) = \begin{cases} n, & \text{if } x \neq n, \\ 0, & \text{otherwise,} \end{cases}$$

for all $x \in L_n$. Imposing the functional equation of $(\text{CP}(N))$ for each discrete negation and distinguishing different cases according to the position of $x, y \in L_n$, we have that there is only one discrete implication that satisfies $(\text{CP}(N_{D_1}))$, and it is given by

$$I_1(x, y) = \begin{cases} 0, & \text{if } y = 0 \text{ and } 1 \leq x \leq n, \\ n, & \text{otherwise,} \end{cases}$$

for all $x, y \in L_n$. Similarly, we find that there is only one discrete implication that satisfies $(\text{CP}(N_{D_2}))$, and it is given by

$$I_2(x, y) = \begin{cases} 0, & \text{if } x = n \text{ and } 0 \leq y \leq n - 1, \\ n, & \text{otherwise,} \end{cases}$$

for all $x, y \in L_n$. Beyond these cases, to date, there is no formula for $|I_n^{\text{CP}}|$ in general, i.e., the number of discrete implications satisfying $(\text{CP}(N))$ with respect to some discrete negation $N$, nor is there a computationally obtained enumeration of it.

## 4.2. Discrete implications satisfying $(\text{NP})$

Nachtegael and Kerre [6] conducted a computational analysis to determine the number of discrete implications satisfying $(\text{NP})$, denoted as $I_n^{\text{NP}}$, for $n \leq 6$. Subsequently, Munar et al. in [19] introduced a recursive formula to calculate $|I_n^{\text{NP}}|$ for any value of $n$. This approach also relied on plane partitions, specifically on those called $(\alpha, \beta)$-monotonic plane partitions, which were initially proposed by Krattenthaler in [65, 66]. These plane partitions can be described as those satisfying $\pi_{i,j} \geq \pi_{i,j+1} + \alpha$ and $\pi_{i,j} \geq \pi_{i+1,j} + \beta$ (following the same nomenclature as in Definition 3.4) whenever both sides are defined. Krattenthaler [65]
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

also provided a closed formula for enumerating $(\alpha, \beta)$-plane partitions whose components of each column are upper-bounded by the corresponding entry of a decreasing $c$-tuple $v$, where $c$ is equal to the number of columns. Although this formula seems adequate at first sight to count these plane partitions, even setting $\alpha = 1$ and $v = (n, n-1, \dots, 2, 1)$ does not give the desired count, since this combination of parameters determines the number of plane partitions whose first row is bounded component-wise by $v$, and not the plane partitions whose first row is equal to $v$, which is the one we are looking for. Consequently, Munar et al. extended this formula and devised a formula for (1, 1)-monotonic plane partitions that the first row is equal to the identity. This formula centers around two operators, denoted as $\Theta$ and $\Upsilon$, which are defined as follows. Given a decreasing $n$-tuple $v = (v_1, \dots, v_n) \in L_n^n$:

* $\Theta(v)$ is the number of binary discrete operators $F: (L_n \setminus \{0\})^2 \to L_n$ which are increasing in each argument and $F(n, y) \le v_{n-y+1}$ for all $y \in L_n \setminus \{0\}$.

* $\Upsilon(v)$ is the number of binary discrete operators $F: (L_n \setminus \{0\})^2 \to L_n$ which are increasing in each argument and $F(n, y) = v_{n-y+1}$ for all $y \in L_n \setminus \{0\}$.

That said, it only remains to calculate $\Theta(v)$ and $\Upsilon(v)$ for any decreasing $n$-tuple $v$.

**Proposition 4.2.** Let $v \in L_n^n$ be a decreasing $n$-tuple. Then, the following statements hold:

* ([19, Proposition 5.2]) $\Theta(v) = \det A(v)$, where $A(v)$ is the $n \times n$ matrix whose entries are

$$A(v)_{i,j} = \binom{n+v_j}{n-i+j},$$

for all $i, j \in \{1, \dots, n\}$.

* ([19, Proposition 5.3]) If $v$ is reduced, i.e., $v = (k, 0, \dots, 0) \in L_n^n$ for some $k \in L_n$, then

$$\Upsilon(v) = \binom{n+k-1}{k}.$$

* [19, Section 5.1] The general formula of $\Upsilon(v)$ is

$$\Upsilon(v) = \begin{cases} \binom{n+v_1-1}{v_1}, & \text{if } v \text{ is reduced,} \\ \Theta(v) - \sum_{x \in V_n(v) \setminus v} \Upsilon(x), & \text{otherwise,} \end{cases}$$

$$\text{where } V_n(v) = \{ \mathbf{x} \in L_n^n \mid x_i \le v_i \text{ and } x_i \ge x_{i+1} \text{ for all } i \in \{1, \dots, n\} \}.$$

Clearly, the formula being sought for $|\mathcal{I}_n^{\text{NP}}|$ is the operator $\Upsilon$ evaluated on the $n$-tuple equal to the identity. Table 8 summarizes some of its values, and we note that since it is a recursive formula it has a high computational cost and therefore it has only been listed for $n \le 10$.

**Proposition 4.3.** ([19, Proposition 5.5]) Let $n \ge 1$ and let us denote by $\mathcal{I}_n^{\text{NP}}$ the set of discrete implications defined on $L_n$ satisfying (NP). Then,

$$|\mathcal{I}_n^{\text{NP}}| = \Upsilon((n, n-1, \dots, 2, 1)). \qquad (51)$$

**Remark 4.1.** In the definition of $\Theta$ and $\Upsilon$, it is stated that they correspond to the enumeration of a certain type of increasing operators in both arguments. This is because in [19] the authors actually exploited the bijection between discrete implications and discrete conjunctions to determine this formula. Indeed, the enumeration of Proposition 4.3 is also true for discrete conjunctions that have neutral element $n$ only in the first argument [19, Proposition 5.1].

**Remark 4.2.** It is indeed remarkable that the computation of $\Theta(v)$, where $v \in L_n^n$, is achieved through the use of a determinant. This approach is a direct consequence of the Lindström-Gessel-Viennot theorem, which was independently proven by Lindström in [67] and by Gessel and Viennot in [68]. This theorem states that the number of paths within a directed graph (a technique commonly employed for counting plane partitions) is equal to the determinant of a matrix. The entries of this matrix correspond to enumerations of simpler cases within this counting process.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

also provided a closed formula for enumerating $(\alpha, \beta)$-plane partitions whose components of each column are upper-bounded by the corresponding entry of a decreasing $c$-tuple $v$, where $c$ is equal to the number of columns. Although this formula seems adequate at first sight to count these plane partitions, even setting $\alpha = 1$ and $v = (n, n-1, \dots, 2, 1)$ does not give the desired count, since this combination of parameters determines the number of plane partitions whose first row is bounded component-wise by $v$, and not the plane partitions whose first row is equal to $v$, which is the one we are looking for. Consequently, Munar et al. extended this formula and devised a formula for (1, 1)-monotonic plane partitions that the first row is equal to the identity. This formula centers around two operators, denoted as $\Theta$ and $\Upsilon$, which are defined as follows. Given a decreasing $n$-tuple $v = (v_1, \dots, v_n) \in L_n^n$:

* $\Theta(v)$ is the number of binary discrete operators $F: (L_n \setminus \{0\})^2 \to L_n$ which are increasing in each argument and $F(n, y) \le v_{n-y+1}$ for all $y \in L_n \setminus \{0\}$.

* $\Upsilon(v)$ is the number of binary discrete operators $F: (L_n \setminus \{0\})^2 \to L_n$ which are increasing in each argument and $F(n, y) = v_{n-y+1}$ for all $y \in L_n \setminus \{0\}$.

That said, it only remains to calculate $\Theta(v)$ and $\Upsilon(v)$ for any decreasing $n$-tuple $v$.

**Proposition 4.2.** Let $v \in L_n^n$ be a decreasing $n$-tuple. Then, the following statements hold:

* ([19, Proposition 5.2]) $\Theta(v) = \det A(v)$, where $A(v)$ is the $n \times n$ matrix whose entries are

$$A(v)_{i,j} = \binom{n+v_j}{n-i+j},$$

for all $i, j \in \{1, \dots, n\}$.

* ([19, Proposition 5.3]) If $v$ is reduced, i.e., $v = (k, 0, \dots, 0) \in L_n^n$ for some $k \in L_n$, then

$$\Upsilon(v) = \binom{n+k-1}{k}.$$

* [19, Section 5.1] The general formula of $\Upsilon(v)$ is

$$\Upsilon(v) = \begin{cases} \binom{n+v_1-1}{v_1}, & \text{if } v \text{ is reduced,} \\ \Theta(v) - \sum_{x \in V_n(v) \setminus v} \Upsilon(x), & \text{otherwise,} \end{cases}$$

where $V_n(v) = \{x \in L_n^n \mid x_i \le v_i \text{ and } x_i \ge x_{i+1} \text{ for all } i \in \{1, \dots, n\} \}$.

Clearly, the formula being sought for $|\mathcal{I}_n^{\text{NP}}|$ is the operator $\Upsilon$ evaluated on the $n$-tuple equal to the identity. Table 8 summarizes some of its values, and we note that since it is a recursive formula it has a high computational cost and therefore it has only been listed for $n \le 10$.

**Proposition 4.3.** ([19, Proposition 5.5]) Let $n \ge 1$ and let us denote by $\mathcal{I}_n^{\text{NP}}$ the set of discrete implications defined on $L_n$ satisfying (NP). Then,

$$|\mathcal{I}_n^{\text{NP}}| = \Upsilon((n, n-1, \dots, 2, 1)). \qquad (51)$$

**Remark 4.1.** In the definition of $\Theta$ and $\Upsilon$, it is stated that they correspond to the enumeration of a certain type of increasing operators in both arguments. This is because in [19] the authors actually exploited the bijection between discrete implications and discrete conjunctions to determine this formula. Indeed, the enumeration of Proposition 4.3 is also true for discrete conjunctions that have neutral element $n$ only in the first argument [19, Proposition 5.1].

**Remark 4.2.** It is indeed remarkable that the computation of $\Theta(v)$, where $v \in L_n^n$, is achieved through the use of a determinant. This approach is a direct consequence of the Lindström-Gessel-Viennot theorem, which was independently proven by Lindström in [67] and by Gessel and Viennot in [68]. This theorem states that the number of paths within a directed graph (a technique commonly employed for counting plane partitions) is equal to the determinant of a matrix. The entries of this matrix correspond to enumerations of simpler cases within this counting process.
### 4.3. Discrete implications satisfying (CB)

For discrete implications satisfying the consequent boundary, $I_n^{CB}$, Nachtegael and Kerre [6] determined their cardinality for $n \le 5$. Similarly, the authors of [19] identified that from the operators $\Theta$ and $\Upsilon$, introduced in Section 4.2, it is possible to determine a general formula for $|I_n^{CB}|$: instead of evaluating $\Upsilon$ only on $(n, n-1, ..., 2, 1)$, it is evaluated on all decreasing $n$-tuples $v$ satisfying $v_i \le i$, and all partial counts are summed. The following proposition establishes this count. As in the enumeration of (NP), for (CB) the proposed formula is also recursive with a high computational cost, so that in Table 8 its values have been illustrated for $n \le 9$.

**Proposition 4.4.** ([19, Proposition 5.6]) Let $n \ge 1$ and let us denote by $I_n^{CB}$ the set of discrete implications defined on $L_n$ satisfying (CB). Then,

$$|I_n^{CB}| = \sum_{v \in \tilde{V}_n} \Upsilon(v), \qquad (52)$$

where $\tilde{V}_n = \{v \in L_n^n | v_1 = n, v_n \le 1, v_i \le n - i + 1 \text{ and } v_i \ge v_{i+1} \text{ for all } 1 \le i \le n - 1\}$.

**Remark 4.3.** Similar to Remark 4.1, the formula of Proposition 4.4 also applies for the enumeration of those discrete conjunctions satisfying $C(x, y) \le y$ for all $x, y \in L_n$, as stated in [19, Proposition 5.1].

**Remark 4.4.** It is trivial to show that if a discrete implication satisfies (NP), then it also satisfies (CB), so that $I_n^{NP} \subseteq I_n^{CB}$. This is confirmed in the enumeration shown in Table 8, since $|I_n^{NP}| \le |I_n^{CB}|$ for all $n \ge 1$.

### 4.4. Discrete implications satisfying (OP)

Setting aside the formulations outlined in Sections 4.2 and 4.3, which are centered on the operators $\Theta$ and $\Upsilon$, and rooted in the framework of $(\alpha, \beta)$-monotonic plane partitions, the authors of [19] also determined the enumeration of discrete implications satisfying (OP), using the so called *staircased plane partitions*, whose origins and their enumeration are also attributed to Krattenthaler in [65]. Nachtegael and Kerre [6] also determined the number of discrete implications that satisfy (OP), for $n \le 6$.

Following the notation in Definition 3.4, a plane partition is staircased when there exist positive and non-zero integers $k$ and $l$ such that its shape $\Lambda = (\lambda_1, ..., \lambda_r)$ satisfies $\lambda_i = l - k \cdot i$ for all $1 \le i \le r$. The enumeration proposed by Krattenthaler in [65, Corollary 10] does not directly yield the formula of this class of discrete implications. Therefore, similarly to their approach with conjunctions, the authors of [19] decomposed $I_n^{OP}$ into distinct subsets, each of which is in bijection with certain sets of staircased plane partitions. The resulting formula they derived is given in the following proposition.

**Proposition 4.5.** ([19, Proposition 5.7]) Let $n \ge 1$, and let us denote by $I_n^{OP}$ the set of discrete implications defined on $L_n$ satisfying (OP). Then,

$$|I_n^{OP}| = \left( \left( \prod_{i=1}^{n} \frac{(2n-i)!}{(n-2+i)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n+1-i+2j}{2} \right) - \left( \prod_{i=1}^{n} \frac{(2n-i-1)!}{(n-3+i)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n-1-i+2j}{2} \right) \right) \\ \cdot \left( 2^{\frac{(n)(n-1)}{2}} \right) \left( \prod_{i=1}^{n-1} i^{n-i} \right) \left( \prod_{i=1}^{n} \frac{1}{(2n+1-2i)!} \right) \qquad (53)$$

for all $n \ge 2$, with $|I_1^{OP}| = 1$.

**Remark 4.5.** As stated in [19, Proposition 5.1], the formula of Proposition 4.5 for $|I_n^{OP}|$ is applicable to enumerate those conjunctions that satisfy $C(x, y) = 0$ if, and only if, $y \le n - x$, with $x, y \in L_n$.

### 4.5. Discrete implications satisfying (IP)

In this section, we present the formula proposed in [19] for the enumeration of discrete implications satisfying (IP), which is also founded on the concept of staircased plane partitions. Similar to the preceding sections, Nachtegael and Kerre computationally determined its cardinality for $n \le 5$.
### 4.3. Discrete implications satisfying (CB)

For discrete implications satisfying the consequent boundary, $I_n^{CB}$, Nachtegael and Kerre [6] determined their cardinality for $n \le 5$. Similarly, the authors of [19] identified that from the operators $\Theta$ and $\Upsilon$, introduced in Section 4.2, it is possible to determine a general formula for $|I_n^{CB}|$: instead of evaluating $\Upsilon$ only on $(n, n-1, \dots, 2, 1)$, it is evaluated on all decreasing $n$-tuples $v$ satisfying $v_i \le i$, and all partial counts are summed. The following proposition establishes this count. As in the enumeration of (NP), for (CB) the proposed formula is also recursive with a high computational cost, so that in Table 8 its values have been illustrated for $n \le 9$.

**Proposition 4.4.** ([19, Proposition 5.6]) Let $n \ge 1$ and let us denote by $I_n^{CB}$ the set of discrete implications defined on $L_n$ satisfying (CB). Then,

$$|I_n^{CB}| = \sum_{v \in \tilde{V}_n} \Upsilon(v), \qquad (52)$$

where $\tilde{V}_n = \{v \in L_n^n | v_1 = n, v_n \le 1, v_i \le n - i + 1 \text{ and } v_i \ge v_{i+1} \text{ for all } 1 \le i \le n - 1\}$.

**Remark 4.3.** Similar to Remark 4.1, the formula of Proposition 4.4 also applies for the enumeration of those discrete conjunctions satisfying $C(x, y) \le y$ for all $x, y \in L_n$, as stated in [19, Proposition 5.1].

**Remark 4.4.** It is trivial to show that if a discrete implication satisfies (NP), then it also satisfies (CB), so that $I_n^{NP} \subseteq I_n^{CB}$. This is confirmed in the enumeration shown in Table 8, since $|I_n^{NP}| \le |I_n^{CB}|$ for all $n \ge 1$.

### 4.4. Discrete implications satisfying (OP)

Setting aside the formulations outlined in Sections 4.2 and 4.3, which are centered on the operators $\Theta$ and $\Upsilon$, and rooted in the framework of $(\alpha, \beta)$-monotonic plane partitions, the authors of [19] also determined the enumeration of discrete implications satisfying (OP), using the so called *staircased plane partitions*, whose origins and their enumeration are also attributed to Krattenthaler in [65]. Nachtegael and Kerre [6] also determined the number of discrete implications that satisfy (OP), for $n \le 6$.

Following the notation in Definition 3.4, a plane partition is staircased when there exist positive and non-zero integers $k$ and $l$ such that its shape $\Lambda = (\lambda_1, \dots, \lambda_r)$ satisfies $\lambda_i = l - k \cdot i$ for all $1 \le i \le r$. The enumeration proposed by Krattenthaler in [65, Corollary 10] does not directly yield the formula of this class of discrete implications. Therefore, similarly to their approach with conjunctions, the authors of [19] decomposed $I_n^{OP}$ into distinct subsets, each of which is in bijection with certain sets of staircased plane partitions. The resulting formula they derived is given in the following proposition.

**Proposition 4.5.** ([19, Proposition 5.7]) Let $n \ge 1$, and let us denote by $I_n^{OP}$ the set of discrete implications defined on $L_n$ satisfying (OP). Then,

$$|I_n^{OP}| = \left( \left( \prod_{i=1}^{n} \frac{(2n-i)!}{(n-2+i)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n+1-i+2j}{2} \right) - \left( \prod_{i=1}^{n} \frac{(2n-i-1)!}{(n-3+i)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n-1-i+2j}{2} \right) \right) \\ \cdot \left( 2^{\frac{(n)(n-1)}{2}} \right) \left( \prod_{i=1}^{n-1} i^{n-i} \right) \left( \prod_{i=1}^{n} \frac{1}{(2n+1-2i)!} \right) \qquad (53)$$

for all $n \ge 2$, with $|I_1^{OP}| = 1$.

**Remark 4.5.** As stated in [19, Proposition 5.1], the formula of Proposition 4.5 for $|I_n^{OP}|$ is applicable to enumerate those conjunctions that satisfy $C(x, y) = 0$ if, and only if, $y \le n - x$, with $x, y \in L_n$.

### 4.5. Discrete implications satisfying (IP)

In this section, we present the formula proposed in [19] for the enumeration of discrete implications satisfying (IP), which is also founded on the concept of staircased plane partitions. Similar to the preceding sections, Nachtegael and Kerre computationally determined its cardinality for $n \le 5$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 8**</caption><thead><tr><th>n</th><th>|I<sub>n</sub>|</th><th>|I<sub>n</sub><sup>SMT</sup>|</th><th>|I<sub>n</sub><sup>CP(N<sub>C</sub>)</sup>|</th><th>|I<sub>n</sub><sup>NP</sup>|</th><th>|I<sub>n</sub><sup>CB</sup>|</th><th>|I<sub>n</sub><sup>OP</sup>|</th><th>|I<sub>n</sub><sup>IP</sup>|</th></tr></thead><tbody><tr><td>1</td><td>= 1 &middot; 10<sup>0</sup></td><td>= 1 &middot; 10<sup>0</sup></td><td>= 1 &middot; 10<sup>0</sup></td><td>= 1 &middot; 10<sup>0</sup></td><td>= 1 &middot; 10<sup>0</sup></td><td>= 1 &middot; 10<sup>0</sup></td><td>= 1 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>= 1.4 &middot; 10<sup>1</sup></td><td>= 2 &middot; 10<sup>0</sup></td><td>= 6 &middot; 10<sup>0</sup></td><td>= 5 &middot; 10<sup>0</sup></td><td>= 8 &middot; 10<sup>0</sup></td><td>= 4 &middot; 10<sup>0</sup></td><td>= 9 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>= 8.05 &middot; 10<sup>2</sup></td><td>= 7 &middot; 10<sup>0</sup></td><td>= 7.7 &middot; 10<sup>1</sup></td><td>= 8.4 &middot; 10<sup>1</sup></td><td>= 2.05 &middot; 10<sup>2</sup></td><td>= 7 &middot; 10<sup>1</sup></td><td>= 2.46 &middot; 10<sup>2</sup></td></tr><tr><td>4</td><td>≈ 2.0815 &middot; 10<sup>5</sup></td><td>= 4.2 &middot; 10<sup>1</sup></td><td>= 2.1 &middot; 10<sup>3</sup></td><td>= 4.719 &middot; 10<sup>3</sup></td><td>= 1.7108 &middot; 10<sup>4</sup></td><td>= 4.125 &middot; 10<sup>3</sup></td><td>= 2.1307 &middot; 10<sup>4</sup></td></tr><tr><td>5</td><td>≈ 2.5041 &middot; 10<sup>8</sup></td><td>= 4.29 &middot; 10<sup>2</sup></td><td>≈ 1.2269 &middot; 10<sup>5</sup></td><td>≈ 8.8488 &middot; 10<sup>5</sup></td><td>≈ 4.6936 &middot; 10<sup>6</sup></td><td>≈ 8.0309 &middot; 10<sup>5</sup></td><td>≈ 5.9679 &middot; 10<sup>6</sup></td></tr><tr><td>6</td><td>≈ 1.4234 &middot; 10<sup>12</sup></td><td>= 7.436 &middot; 10<sup>3</sup></td><td>≈ 1.5459 &middot; 10<sup>7</sup></td><td>≈ 5.5336 &middot; 10<sup>8</sup></td><td>≈ 4.2538 &middot; 10<sup>9</sup></td><td>≈ 5.1624 &middot; 10<sup>8</sup></td><td>≈ 5.4648 &middot; 10<sup>9</sup></td></tr><tr><td>7</td><td>≈ 3.8534 &middot; 10<sup>16</sup></td><td>≈ 2.1835 &middot; 10<sup>5</sup></td><td>≈ 4.2204 &middot; 10<sup>9</sup></td><td>≈ 1.1535 &middot; 10<sup>12</sup></td><td>≈ 1.2769 &middot; 10<sup>13</sup></td><td>≈ 1.0977 &middot; 10<sup>12</sup></td><td>≈ 1.6465 &middot; 10<sup>13</sup></td></tr><tr><td>8</td><td>≈ 4.9888 &middot; 10<sup>21</sup></td><td>≈ 1.085 &middot; 10<sup>7</sup></td><td>≈ 2.5045 &middot; 10<sup>12</sup></td><td>≈ 8.0122 &middot; 10<sup>15</sup></td><td>≈ 1.2715 &middot; 10<sup>17</sup></td><td>≈ 7.7337 &middot; 10<sup>15</sup></td><td>≈ 1.6387 &middot; 10<sup>17</sup></td></tr><tr><td>9</td><td>≈ 3.0961 &middot; 10<sup>27</sup></td><td>≈ 9.1184 &middot; 10<sup>8</sup></td><td>≈ 3.238 &middot; 10<sup>15</sup></td><td>≈ 1.8542 &middot; 10<sup>20</sup></td><td>≈ 4.2044 &middot; 10<sup>21</sup></td><td>≈ 1.808 &middot; 10<sup>20</sup></td><td>≈ 5.4014 &middot; 10<sup>21</sup></td></tr><tr><td>10</td><td>≈ 9.2223 &middot; 10<sup>33</sup></td><td>≈ 1.2953 &middot; 10<sup>11</sup></td><td>≈ 9.1352 &middot; 10<sup>18</sup></td><td>≈ 4.6196 &middot; 10<sup>26</sup></td><td>—</td><td>≈ 1.404 &middot; 10<sup>25</sup></td><td>≈ 5.9067 &middot; 10<sup>26</sup></td></tr><tr><td>11</td><td>≈ 1.3195 &middot; 10<sup>41</sup></td><td>≈ 3.1096 &middot; 10<sup>13</sup></td><td>≈ 5.6307 &middot; 10<sup>22</sup></td><td>—</td><td>—</td><td>≈ 3.6241 &middot; 10<sup>30</sup></td><td>≈ 2.1453 &middot; 10<sup>32</sup></td></tr><tr><td>12</td><td>≈ 9.0708 &middot; 10<sup>48</sup></td><td>≈ 1.2611 &middot; 10<sup>16</sup></td><td>≈ 7.5891 &middot; 10<sup>26</sup></td><td>—</td><td>—</td><td>≈ 3.1111 &middot; 10<sup>36</sup></td><td>≈ 2.59 &middot; 10<sup>38</sup></td></tr><tr><td>13</td><td>≈ 2.997 &middot; 10<sup>57</sup></td><td>≈ 8.6394 &middot; 10<sup>18</sup></td><td>≈ 2.2381 &middot; 10<sup>31</sup></td><td>—</td><td>—</td><td>≈ 8.8857 &middot; 10<sup>42</sup></td><td>≈ 1.0399 &middot; 10<sup>45</sup></td></tr></tbody></table>

Enumeration of some classes of discrete implications, for $n \le 13$. From left to right: discrete implications in general, given by Proposition 3.5; smooth discrete implications, given by Proposition 3.9; discrete implications satisfying ($CP(N_c)$), given by Proposition 3.6; discrete implications satisfying (NP), given by Proposition 4.3; discrete implications satisfying (CB), given by Proposition 4.4; discrete implications satisfying (OP), given by Proposition 4.5; and finally discrete implications satisfying (IP), given by Proposition 4.6.

**Proposition 4.6.** ([19, Proposition 5.8]) Let $n \ge 1$, and let us denote by $I_n^{IP}$ the set of discrete implications defined on $L_n$ satisfying (IP). Then,

$$|I_n^{IP}| = \left( \left( \prod_{i=1}^{n} \frac{(2n-i+1)!}{(n-i-1)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n+3-i+2j}{2} \right) - \left( \prod_{i=1}^{n} \frac{(2n-i)!}{(n-2+i)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n+1-i+2j}{2} \right) \right) \cdot \left( 2^{\frac{(n)(n-1)}{2}} \right) \left( \prod_{i=1}^{n-1} i^{n-i} \right) \left( \prod_{i=1}^{n} \frac{1}{(2n+1-2i)!} \right) \quad (54)$$

for all $n \ge 2$, with $|I_1^{IP}| = 1$.

**Remark 4.6.** As stated in [19, Proposition 5.1], the formula of Proposition 4.6 for $|I_n^{IP}|$ is applicable to enumerate those conjunctions that satisfy $C(x, y) = 0$ for all $y \le n - x$, with $x, y \in L_n$.

**Remark 4.7.** Remark 4.4 shows that the relationship between (NP) and (CB) is observed through the enumeration of both classes of discrete implications. Similarly, this is also the case for (OP) and (IP). If a discrete implication satisfies (OP), it also satisfies (IP), so $I_n^{OP} \subseteq I_n^{IP}$ and consequently $|I_n^{OP}| \le |I_n^{IP}|$ for all $n \ge 1$, as it can be shown in Table 8.

# 5. New formulas for the enumeration of some classes of operators

In this section, we further explore the enumerations of operators on $L_n$, presenting some novel formulas. In Section 5.1, we prove the enumeration of smooth discrete aggregation functions with a neutral element, or with an absorbing element. In Section 5.2, we present the enumeration of idempotent discrete conjunctions and commutative idempotent discrete conjunctions.

## 5.1. Enumeration of smooth discrete aggregation functions having neutral or absorbing element

In this section, we focus on the enumeration of two classes of smooth discrete aggregation functions: those with a neutral element, and those with an absorbing element. The proposed formulas are constructed from the existing characterizations of these discrete aggregation functions, proposed by Mas et al. in [9].
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 8**</caption><thead><tr><th>n</th><th>|I<sub>n</sub>|</th><th>|I<sub>n</sub><sup>SMT</sup>|</th><th>|I<sub>n</sub><sup>CP(N<sub>C</sub>)</sup>|</th><th>|I<sub>n</sub><sup>NP</sup>|</th><th>|I<sub>n</sub><sup>CB</sup>|</th><th>|I<sub>n</sub><sup>OP</sup>|</th><th>|I<sub>n</sub><sup>IP</sup>|</th></tr></thead><tbody><tr><td>1</td><td>= 1 · 10<sup>0</sup></td><td>= 1 · 10<sup>0</sup></td><td>= 1 · 10<sup>0</sup></td><td>= 1 · 10<sup>0</sup></td><td>= 1 · 10<sup>0</sup></td><td>= 1 · 10<sup>0</sup></td><td>= 1 · 10<sup>0</sup></td></tr><tr><td>2</td><td>= 1.4 · 10<sup>1</sup></td><td>= 2 · 10<sup>0</sup></td><td>= 6 · 10<sup>0</sup></td><td>= 5 · 10<sup>0</sup></td><td>= 8 · 10<sup>0</sup></td><td>= 4 · 10<sup>0</sup></td><td>= 9 · 10<sup>0</sup></td></tr><tr><td>3</td><td>= 8.05 · 10<sup>2</sup></td><td>= 7 · 10<sup>0</sup></td><td>= 7.7 · 10<sup>1</sup></td><td>= 8.4 · 10<sup>1</sup></td><td>= 2.05 · 10<sup>2</sup></td><td>= 7 · 10<sup>1</sup></td><td>= 2.46 · 10<sup>2</sup></td></tr><tr><td>4</td><td>≈ 2.0815 · 10<sup>5</sup></td><td>= 4.2 · 10<sup>1</sup></td><td>= 2.1 · 10<sup>3</sup></td><td>= 4.719 · 10<sup>3</sup></td><td>= 1.7108 · 10<sup>4</sup></td><td>= 4.125 · 10<sup>3</sup></td><td>= 2.1307 · 10<sup>4</sup></td></tr><tr><td>5</td><td>≈ 2.5041 · 10<sup>8</sup></td><td>= 4.29 · 10<sup>2</sup></td><td>≈ 1.2269 · 10<sup>5</sup></td><td>≈ 8.8488 · 10<sup>5</sup></td><td>≈ 4.6936 · 10<sup>6</sup></td><td>≈ 8.0309 · 10<sup>5</sup></td><td>≈ 5.9679 · 10<sup>6</sup></td></tr><tr><td>6</td><td>≈ 1.4234 · 10<sup>12</sup></td><td>= 7.436 · 10<sup>3</sup></td><td>≈ 1.5459 · 10<sup>7</sup></td><td>≈ 5.5336 · 10<sup>8</sup></td><td>≈ 4.2538 · 10<sup>9</sup></td><td>≈ 5.1624 · 10<sup>8</sup></td><td>≈ 5.4648 · 10<sup>9</sup></td></tr><tr><td>7</td><td>≈ 3.8534 · 10<sup>16</sup></td><td>≈ 2.1835 · 10<sup>5</sup></td><td>≈ 4.2204 · 10<sup>9</sup></td><td>≈ 1.1535 · 10<sup>12</sup></td><td>≈ 1.2769 · 10<sup>13</sup></td><td>≈ 1.0977 · 10<sup>12</sup></td><td>≈ 1.6465 · 10<sup>13</sup></td></tr><tr><td>8</td><td>≈ 4.9888 · 10<sup>21</sup></td><td>≈ 1.085 · 10<sup>7</sup></td><td>≈ 2.5045 · 10<sup>12</sup></td><td>≈ 8.0122 · 10<sup>15</sup></td><td>≈ 1.2715 · 10<sup>17</sup></td><td>≈ 7.7337 · 10<sup>15</sup></td><td>≈ 1.6387 · 10<sup>17</sup></td></tr><tr><td>9</td><td>≈ 3.0961 · 10<sup>27</sup></td><td>≈ 9.1184 · 10<sup>8</sup></td><td>≈ 3.238 · 10<sup>15</sup></td><td>≈ 1.8542 · 10<sup>20</sup></td><td>≈ 4.2044 · 10<sup>21</sup></td><td>≈ 1.808 · 10<sup>20</sup></td><td>≈ 5.4014 · 10<sup>21</sup></td></tr><tr><td>10</td><td>≈ 9.2223 · 10<sup>33</sup></td><td>≈ 1.2953 · 10<sup>11</sup></td><td>≈ 9.1352 · 10<sup>18</sup></td><td>≈ 4.6196 · 10<sup>26</sup></td><td>—</td><td>≈ 1.404 · 10<sup>25</sup></td><td>≈ 5.9067 · 10<sup>26</sup></td></tr><tr><td>11</td><td>≈ 1.3195 · 10<sup>41</sup></td><td>≈ 3.1096 · 10<sup>13</sup></td><td>≈ 5.6307 · 10<sup>22</sup></td><td>—</td><td>—</td><td>≈ 3.6241 · 10<sup>30</sup></td><td>≈ 2.1453 · 10<sup>32</sup></td></tr><tr><td>12</td><td>≈ 9.0708 · 10<sup>48</sup></td><td>≈ 1.2611 · 10<sup>16</sup></td><td>≈ 7.5891 · 10<sup>26</sup></td><td>—</td><td>—</td><td>≈ 3.1111 · 10<sup>36</sup></td><td>≈ 2.59 · 10<sup>38</sup></td></tr><tr><td>13</td><td>≈ 2.997 · 10<sup>57</sup></td><td>≈ 8.6394 · 10<sup>18</sup></td><td>≈ 2.2381 · 10<sup>31</sup></td><td>—</td><td>—</td><td>≈ 8.8857 · 10<sup>42</sup></td><td>≈ 1.0399 · 10<sup>45</sup></td></tr></tbody></table>

Enumeration of some classes of discrete implications, for $n \le 13$. From left to right: discrete implications in general, given by Proposition 3.5; smooth discrete implications, given by Proposition 3.9; discrete implications satisfying ($CP(N_c)$), given by Proposition 3.6; discrete implications satisfying (NP), given by Proposition 4.3; discrete implications satisfying (CB), given by Proposition 4.4; discrete implications satisfying (OP), given by Proposition 4.5; and finally discrete implications satisfying (IP), given by Proposition 4.6.

**Proposition 4.6.** ([19, Proposition 5.8]) Let $n \ge 1$, and let us denote by $I_n^{IP}$ the set of discrete implications defined on $L_n$ satisfying (IP). Then,

$$ |I_n^{IP}| = \left( \left( \prod_{i=1}^{n} \frac{(2n-i+1)!}{(n-i-1)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n+3-i+2j}{2} \right) - \left( \prod_{i=1}^{n} \frac{(2n-i)!}{(n-2+i)!} \right) \left( \prod_{i=2}^{n} \prod_{j=0}^{i-2} \frac{3n+1-i+2j}{2} \right) \right) \cdot \left( 2^{\frac{(n)(n-1)}{2}} \right) \left( \prod_{i=1}^{n-1} i^{n-i} \right) \left( \prod_{i=1}^{n} \frac{1}{(2n+1-2i)!} \right) \quad (54) $$

for all $n \ge 2$, with $|I_1^{IP}| = 1$.

**Remark 4.6.** As stated in [19, Proposition 5.1], the formula of Proposition 4.6 for $|I_n^{IP}|$ is applicable to enumerate those conjunctions that satisfy $C(x, y) = 0$ for all $y \le n - x$, with $x, y \in L_n$.

**Remark 4.7.** Remark 4.4 shows that the relationship between (NP) and (CB) is observed through the enumeration of both classes of discrete implications. Similarly, this is also the case for (OP) and (IP). If a discrete implication satisfies (OP), it also satisfies (IP), so $I_n^{OP} \subseteq I_n^{IP}$ and consequently $|I_n^{OP}| \le |I_n^{IP}|$ for all $n \ge 1$, as it can be shown in Table 8.

# 5. New formulas for the enumeration of some classes of operators

In this section, we further explore the enumerations of operators on $L_n$, presenting some novel formulas. In Section 5.1, we prove the enumeration of smooth discrete aggregation functions with a neutral element, or with an absorbing element. In Section 5.2, we present the enumeration of idempotent discrete conjunctions and commutative idempotent discrete conjunctions.

## 5.1. Enumeration of smooth discrete aggregation functions having neutral or absorbing element

In this section, we focus on the enumeration of two classes of smooth discrete aggregation functions: those with a neutral element, and those with an absorbing element. The proposed formulas are constructed from the existing characterizations of these discrete aggregation functions, proposed by Mas et al. in [9].
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

These results rely on the combination of a discrete quasi-copula (see Section 3.3 for more information), and a discrete dual quasi-copula, which we recall below.

**Definition 5.1.** ([9]) A discrete dual quasi-copula is a discrete disjunction $Q' : L_n^2 \to L_n$ which is 1-Lipschitz and has 0 as neutral element.

Discrete dual quasi-copulas were originally introduced by Mas et al. in [9]; however, the primary focus of their work was to study smooth discrete aggregation functions so an enumeration of discrete dual quasi-copulas is not yet known. It is important to highlight that the main difference between discrete quasi-copulas and discrete dual quasi-copulas lies in the location of the neutral element. This distinction serves as a reminder of the duality observed between discrete conjunctions and discrete disjunctions, and therefore we can derive the following result.

**Proposition 5.1.** Let $n \ge 1$, and let us denote by $Q_n$ and $Q'_n$ the sets of discrete quasi-copulas and discrete dual quasi-copulas, respectively, defined on $L_n$. Let $\varphi : Q_n \to Q'_n$ be defined point-wise by

$$ \varphi(Q)(x, y) = n - Q(n - x, n - y), $$

for all $x, y \in L_n$, where $Q \in Q_n$. Then, $\varphi(Q)$ is a discrete dual quasi-copula, and the transformation $\varphi$ is a bijection.

*Proof.* Let $Q' = \varphi(Q)$. It is straightforward to show that $Q$ is monotone if, and only if, $Q'$ is; $Q$ is 1-Lipschitz if, and only if, $Q'$ is; and finally that $Q$ has neutral element $n$ if, and only if, $Q'$ has neutral element 0. Moreover, $\varphi^{-1}(Q') = n - Q'(n - x, n - y)$, for all $x, y \in L_n$, so $\varphi$ is a bijection. $\square$

We have discussed in Section 3.3 that, actually, discrete quasi-copulas are nothing more than smooth discrete conjunctions. With this result, we can therefore conclude that discrete dual quasi-copulas proposed in [9] are really smooth discrete disjunctions. Then, their enumeration is given by Proposition 3.9. Said that, we reformulate the characterization of smooth discrete aggregation functions with neutral element or absorbing element $\alpha \in L_n$ on the basis of smooth discrete conjunctions and disjunctions.

**Theorem 5.1.** ([9, Theorem 2]) Let $\alpha \in L_n$ and $F : L_n^2 \to L_n$ be a binary operator on $L_n$. Then, the following statements are equivalent:

(i) $F$ is a smooth discrete aggregation function with absorbing element $\alpha$.

(ii) There exist a smooth discrete conjunction $C$ on $[0, \alpha]$ and a smooth discrete disjunction $D$ on $[\alpha, n]$ such that $F$ is given by

$$ F(x, y) = \begin{cases} D(x, y), & \text{if } (x, y) \in [0, \alpha]^2, \\ C(x, y), & \text{if } (x, y) \in [\alpha, n]^2, \\ \alpha, & \text{otherwise,} \end{cases} \qquad (55) $$

for all $x, y \in L_n$.

**Theorem 5.2.** ([9, Theorem 3]) Let $\alpha \in L_n$ and $F : L_n^2 \to L_n$ be a binary operator on $L_n$. Then, the following statements are equivalent:

(i) $F$ is a smooth discrete aggregation function with neutral element $\alpha$.

(ii) There exist a smooth discrete conjunction $C$ on $[0, \alpha]$ and a smooth discrete disjunction $D$ on $[\alpha, n]$ such that $F$ is given by

$$ F(x, y) = \begin{cases} C(x, y), & \text{if } (x, y) \in [0, \alpha]^2, \\ D(x, y), & \text{if } (x, y) \in [\alpha, n]^2, \\ x + y - \alpha, & \text{otherwise,} \end{cases} \qquad (56) $$

for all $x, y \in L_n$.

With these results, we propose the enumeration of discrete aggregation functions that have absorbing element or neutral element $\alpha \in L_n$. Table 9 illustrates the first 13 values of this enumeration.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

These results rely on the combination of a discrete quasi-copula (see Section 3.3 for more information), and a discrete dual quasi-copula, which we recall below.

**Definition 5.1.** ([9]) A discrete dual quasi-copula is a discrete disjunction $Q' : L_n^2 \to L_n$ which is 1-Lipschitz and has 0 as neutral element.

Discrete dual quasi-copulas were originally introduced by Mas et al. in [9]; however, the primary focus of their work was to study smooth discrete aggregation functions so an enumeration of discrete dual quasi-copulas is not yet known. It is important to highlight that the main difference between discrete quasi-copulas and discrete dual quasi-copulas lies in the location of the neutral element. This distinction serves as a reminder of the duality observed between discrete conjunctions and discrete disjunctions, and therefore we can derive the following result.

**Proposition 5.1.** Let $n \ge 1$, and let us denote by $Q_n$ and $Q'_n$ the sets of discrete quasi-copulas and discrete dual quasi-copulas, respectively, defined on $L_n$. Let $\varphi : Q_n \to Q'_n$ be defined point-wise by

$$ \varphi(Q)(x, y) = n - Q(n - x, n - y), $$

for all $x, y \in L_n$, where $Q \in Q_n$. Then, $\varphi(Q)$ is a discrete dual quasi-copula, and the transformation $\varphi$ is a bijection.

*Proof.* Let $Q' = \varphi(Q)$. It is straightforward to show that $Q$ is monotone if, and only if, $Q'$ is; $Q$ is 1-Lipschitz if, and only if, $Q'$ is; and finally that $Q$ has neutral element $n$ if, and only if, $Q'$ has neutral element 0. Moreover, $\varphi^{-1}(Q') = n - Q'(n - x, n - y)$, for all $x, y \in L_n$, so $\varphi$ is a bijection. $\square$

We have discussed in Section 3.3 that, actually, discrete quasi-copulas are nothing more than smooth discrete conjunctions. With this result, we can therefore conclude that discrete dual quasi-copulas proposed in [9] are really smooth discrete disjunctions. Then, their enumeration is given by Proposition 3.9. Said that, we reformulate the characterization of smooth discrete aggregation functions with neutral element or absorbing element $\alpha \in L_n$ on the basis of smooth discrete conjunctions and disjunctions.

**Theorem 5.1.** ([9, Theorem 2]) Let $\alpha \in L_n$ and $F : L_n^2 \to L_n$ be a binary operator on $L_n$. Then, the following statements are equivalent:

(i) $F$ is a smooth discrete aggregation function with absorbing element $\alpha$.

(ii) There exist a smooth discrete conjunction $C$ on $[0, \alpha]$ and a smooth discrete disjunction $D$ on $[\alpha, n]$ such that $F$ is given by

$$ F(x, y) = \begin{cases} D(x, y), & \text{if } (x, y) \in [0, \alpha]^2, \\ C(x, y), & \text{if } (x, y) \in [\alpha, n]^2, \\ \alpha, & \text{otherwise,} \end{cases} \qquad (55) $$

for all $x, y \in L_n$.

**Theorem 5.2.** ([9, Theorem 3]) Let $\alpha \in L_n$ and $F : L_n^2 \to L_n$ be a binary operator on $L_n$. Then, the following statements are equivalent:

(i) $F$ is a smooth discrete aggregation function with neutral element $\alpha$.

(ii) There exist a smooth discrete conjunction $C$ on $[0, \alpha]$ and a smooth discrete disjunction $D$ on $[\alpha, n]$ such that $F$ is given by

$$ F(x, y) = \begin{cases} C(x, y), & \text{if } (x, y) \in [0, \alpha]^2, \\ D(x, y), & \text{if } (x, y) \in [\alpha, n]^2, \\ x + y - \alpha, & \text{otherwise,} \end{cases} \qquad (56) $$

for all $x, y \in L_n$.

With these results, we propose the enumeration of discrete aggregation functions that have absorbing element or neutral element $\alpha \in L_n$. Table 9 illustrates the first 13 values of this enumeration.
**Proposition 5.2.** For all $n \ge 2$, the following statements hold:

(i) The number $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|$ of smooth discrete aggregation functions on $L_n$ with absorbing element $\alpha \in L_n \setminus \{0, n\}$, coincides with the number $|A_n^{\text{SMT}} \cap A_n^{\text{NE}(\alpha)}|$ of smooth discrete aggregation functions on $L_n$ with neutral element $\alpha \in L_n \setminus \{0, n\}$, and are given by

$$|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}| = |A_n^{\text{SMT}} \cap A_n^{\text{NE}(\alpha)}| = \left( \prod_{i=0}^{\alpha-1} \frac{(3i+1)!}{(\alpha+i)!} \right) \left( \prod_{i=0}^{n-\alpha-1} \frac{(3i+1)!}{(n-\alpha+i)!} \right). \quad (57)$$

(ii) The number $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}}|$ of smooth discrete aggregation functions on $L_n$ with absorbing element, coincides with the number $|A_n^{\text{SMT}} \cap A_n^{\text{NE}}|$ of smooth discrete aggregation functions on $L_n$ with neutral element, and are given by

$$|A_n^{\text{SMT}} \cap A_n^{\text{ABS}}| = |A_n^{\text{SMT}} \cap A_n^{\text{NE}}| = 2 \cdot \prod_{i=0}^{n-1} \frac{(3i+1)!}{(n+i)!} + \sum_{\alpha=1}^{n-1} \left( \prod_{i=0}^{\alpha-1} \frac{(3i+1)!}{(\alpha+i)!} \right) \left( \prod_{i=0}^{n-\alpha-1} \frac{(3i+1)!}{(n-\alpha+i)!} \right). \quad (58)$$

*Proof.* Let us prove each statement. Regarding the first one, we prove the case for the absorbing element $\alpha$. Using the characterization of Theorem 5.1, given $\alpha \in L_n \setminus \{0, n\}$, the number of smooth discrete aggregation functions with absorbing element $\alpha$ is given by the product of the number of smooth discrete disjunctions that can be constructed on $[0, \alpha]$, and the number of smooth discrete conjunctions that can be constructed on $[\alpha, n]$. The first of these two enumerations is straightforward, and is given by Proposition 3.9. Regarding the second, let us observe that a binary operator $C$ is a discrete conjunction on an interval $[a, b]$, with $a < b$ if, and only if, there exists a discrete conjunction $C'$ in $L_{b-a}$ such that $C'(x, y) = C(x + a, y + a) - a$, for all $x, y \in L_{b-a}$. Therefore, the number of smooth discrete conjunctions that can be constructed on $[\alpha, n]$ is equal to the number of smooth discrete conjunctions that can be constructed on $L_{n-\alpha}$, which is also given by Proposition 3.9. For the case involving the neutral element $\alpha$, an analogous argument proves the equality.

Let us look at the second statement, and focus on the enumeration of $A_n^{\text{SMT}} \cap A_n^{\text{ABS}}$, which trivially is equal to

$$|A_n^{\text{SMT}} \cap A_n^{\text{ABS}}| = \sum_{\alpha \in L_n} |A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|.$$

When $\alpha = 0$ or $\alpha = n$, the operators of $A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}$ are smooth discrete conjunctions or smooth discrete disjunctions, respectively, whose enumeration is the same and they are given by Proposition 3.9. When $1 \le \alpha \le n-1$, $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|$ is given in the previous statement, and therefore the formula holds. Finally, for the enumeration of $A_n^{\text{SMT}} \cap A_n^{\text{NE}}$, it suffices to reproduce a similar argument. $\square$

**Remark 5.1.** As indicated in the proof of Proposition 5.2, if the absorbing element in statement (i) is $\alpha = 0$ or $\alpha = n$, then the formula of $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|$ coincides with that of Proposition 3.9. The same fact occurs if $\alpha$ is a neutral element.

## 5.2. Enumeration of idempotent discrete conjunctions

In this section, we delve into the realm of idempotent discrete conjunctions. To initiate our exploration, we shall consider an idempotent discrete conjunction $C$ on $L_n$, which can be visually represented by the following diagram:
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Proposition 5.2.** For all $n \ge 2$, the following statements hold:

(i) The number $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|$ of smooth discrete aggregation functions on $L_n$ with absorbing element $\alpha \in L_n \setminus \{0, n\}$, coincides with the number $|A_n^{\text{SMT}} \cap A_n^{\text{NE}(\alpha)}|$ of smooth discrete aggregation functions on $L_n$ with neutral element $\alpha \in L_n \setminus \{0, n\}$, and are given by

$$|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}| = |A_n^{\text{SMT}} \cap A_n^{\text{NE}(\alpha)}| = \left( \prod_{i=0}^{\alpha-1} \frac{(3i+1)!}{(\alpha+i)!} \right) \left( \prod_{i=0}^{n-\alpha-1} \frac{(3i+1)!}{(n-\alpha+i)!} \right). \quad (57)$$

(ii) The number $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}}|$ of smooth discrete aggregation functions on $L_n$ with absorbing element, coincides with the number $|A_n^{\text{SMT}} \cap A_n^{\text{NE}}|$ of smooth discrete aggregation functions on $L_n$ with neutral element, and are given by

$$|A_n^{\text{SMT}} \cap A_n^{\text{ABS}}| = |A_n^{\text{SMT}} \cap A_n^{\text{NE}}| = 2 \cdot \prod_{i=0}^{n-1} \frac{(3i+1)!}{(n+i)!} + \sum_{\alpha=1}^{n-1} \left( \prod_{i=0}^{\alpha-1} \frac{(3i+1)!}{(\alpha+i)!} \right) \left( \prod_{i=0}^{n-\alpha-1} \frac{(3i+1)!}{(n-\alpha+i)!} \right). \quad (58)$$

*Proof.* Let us prove each statement. Regarding the first one, we prove the case for the absorbing element $\alpha$. Using the characterization of Theorem 5.1, given $\alpha \in L_n \setminus \{0, n\}$, the number of smooth discrete aggregation functions with absorbing element $\alpha$ is given by the product of the number of smooth discrete disjunctions that can be constructed on $[0, \alpha]$, and the number of smooth discrete conjunctions that can be constructed on $[\alpha, n]$. The first of these two enumerations is straightforward, and is given by Proposition 3.9. Regarding the second, let us observe that a binary operator $C$ is a discrete conjunction on an interval $[a, b]$, with $a < b$ if, and only if, there exists a discrete conjunction $C'$ in $L_{b-a}$ such that $C'(x, y) = C(x + a, y + a) - a$, for all $x, y \in L_{b-a}$. Therefore, the number of smooth discrete conjunctions that can be constructed on $[\alpha, n]$ is equal to the number of smooth discrete conjunctions that can be constructed on $L_{n-\alpha}$, which is also given by Proposition 3.9. For the case involving the neutral element $\alpha$, an analogous argument proves the equality.

Let us look at the second statement, and focus on the enumeration of $A_n^{\text{SMT}} \cap A_n^{\text{ABS}}$, which trivially is equal to

$$|A_n^{\text{SMT}} \cap A_n^{\text{ABS}}| = \sum_{\alpha \in L_n} |A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|.$$

When $\alpha = 0$ or $\alpha = n$, the operators of $A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}$ are smooth discrete conjunctions or smooth discrete disjunctions, respectively, whose enumeration is the same and they are given by Proposition 3.9. When $1 \le \alpha \le n-1$, $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|$ is given in the previous statement, and therefore the formula holds. Finally, for the enumeration of $A_n^{\text{SMT}} \cap A_n^{\text{NE}}$, it suffices to reproduce a similar argument. $\square$

**Remark 5.1.** As indicated in the proof of Proposition 5.2, if the absorbing element in statement (i) is $\alpha = 0$ or $\alpha = n$, then the formula of $|A_n^{\text{SMT}} \cap A_n^{\text{ABS}(\alpha)}|$ coincides with that of Proposition 3.9. The same fact occurs if $\alpha$ is a neutral element.

## 5.2. Enumeration of idempotent discrete conjunctions

In this section, we delve into the realm of idempotent discrete conjunctions. To initiate our exploration, we shall consider an idempotent discrete conjunction $C$ on $L_n$, which can be visually represented by the following diagram:
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 9**</caption><thead><tr><th>n</th><th>|A<sub>n</sub><sup>SMT</sup> ∩ A<sub>n</sub><sup>ABS</sup>|, |A<sub>n</sub><sup>SMT</sup> ∩ A<sub>n</sub><sup>NE</sup>|</th><th>C<sub>n</sub><sup>COM</sup> ∩ C<sub>n</sub><sup>IDE</sup></th><th>C<sub>n</sub><sup>IDE</sup></th></tr></thead><tbody><tr><td>1</td><td>2 = 2.10<sup>0</sup></td><td>1 = 1.10<sup>0</sup></td><td>1 = 1.10<sup>0</sup></td></tr><tr><td>2</td><td>5 = 5.10<sup>0</sup></td><td>2 = 2.10<sup>0</sup></td><td>4 = 4.10<sup>0</sup></td></tr><tr><td>3</td><td>18 = 1.8.10<sup>1</sup></td><td>8 = 8.10<sup>0</sup></td><td>64 = 6.4.10<sup>1</sup></td></tr><tr><td>4</td><td>102 = 1.02.10<sup>2</sup></td><td>64 = 6.4.10<sup>1</sup></td><td>4,096 = 4.096.10<sup>3</sup></td></tr><tr><td>5</td><td>970 = 9.7.10<sup>2</sup></td><td>1,024 = 1.024.10<sup>3</sup></td><td>≈ 1.0486.10<sup>6</sup></td></tr><tr><td>6</td><td>15,947 = 1.5947.10<sup>4</sup></td><td>32,768 = 3.2768.10<sup>4</sup></td><td>≈ 1.0737.10<sup>9</sup></td></tr><tr><td>7</td><td>≈ 4.5387.10<sup>5</sup></td><td>≈ 2.0972.10<sup>6</sup></td><td>≈ 4.398.10<sup>12</sup></td></tr><tr><td>8</td><td>≈ 2.2175.10<sup>7</sup></td><td>≈ 2.6844.10<sup>8</sup></td><td>≈ 7.2058.10<sup>16</sup></td></tr><tr><td>9</td><td>≈ 1.8464.10<sup>9</sup></td><td>≈ 6.8719.10<sup>10</sup></td><td>≈ 4.7224.10<sup>21</sup></td></tr><tr><td>10</td><td>≈ 2.6094.10<sup>11</sup></td><td>≈ 3.5184.10<sup>13</sup></td><td>≈ 1.2379.10<sup>27</sup></td></tr><tr><td>11</td><td>≈ 6.2454.10<sup>13</sup></td><td>≈ 3.6029.10<sup>16</sup></td><td>≈ 1.2981.10<sup>33</sup></td></tr><tr><td>12</td><td>≈ 2.5285.10<sup>16</sup></td><td>≈ 7.3787.10<sup>19</sup></td><td>≈ 5.4445.10<sup>39</sup></td></tr><tr><td>13</td><td>≈ 1.7304.10<sup>19</sup></td><td>≈ 3.0223.10<sup>23</sup></td><td>≈ 9.1344.10<sup>46</sup></td></tr></tbody></table>

Enumeration of the proposed formulas. From left to right: smooth discrete aggregation function with neutral element or with absorbing element, given by Proposition 5.2; commutative and idempotent discrete conjunctions, and idempotent discrete conjunctions, given by Proposition 5.4.

<figure><img src="image_1.png" /></figure>

Let us represent the set of idempotent discrete conjunctions as $C_n^{IDE}$. Looking at the above diagram, it is possible to decompose any idempotent discrete conjunction $C$ into two commutative idempotent discrete conjunctions. This arises from the fact that values of $C$ located above the diagonal $y = x$, with $x \in L_n$, and those below the diagonal, have no relation and are independent. Consequently, determining the enumeration of $|C_n^{COM} \cap C_n^{IDE}|$ provides a solution to the initial problem. We formalize this insight in the following proposition.

**Proposition 5.3.** Let $n \ge 1$, and let us denote by $C_n^{IDE}$ and $C_n^{COM}$ the sets of idempotent discrete conjunctions and commutative discrete conjunctions, respectively, defined on $L_n$. Let $\varphi : C_n^{IDE} \to (C_n^{COM} \cap C_n^{IDE}) \times (C_n^{COM} \cap C_n^{IDE})$ be defined by $\varphi(C) = (C_1, C_2)$, with $C \in C_n^{IDE}$, $C_1$ and $C_2$ are given point-wise by $C_1(x, y) = C_1(y, x) = C(x, y)$ and $C_2(x, y) = C_2(y, x) = C(y, x)$ for all $x \in L_n$ and $y \in [x, n]$. Then, the transformation $\varphi$ is a bijection.

*Proof.* First, $\varphi$ is well defined since $C_1$ and $C_2$ are commutative and idempotent by construction. Next, the injective and exhaustive nature of $\varphi$ follows directly from the expressions of $C_1$ and $C_2$, since the common points of $C_1$ and $C_2$ are only those on the diagonal of $C$. Then, it is a bijective transformation, and $\varphi^{-1}$ is given by $\varphi^{-1}(C_1, C_2) = C$, where $C(x, y) = C_1(x, y)$ and $C(y, x) = C_2(x, y)$ for all $x \in L_n$ and $y \in [x, n]$. $\square$

With this result, we can observe that $|C_n^{IDE}| = |C_n^{COM} \cap C_n^{IDE}|^2$ for all $n \ge 1$. To find the enumeration of $C_n^{COM} \cap C_n^{IDE}$, let us consider the values of any commutative and idempotent discrete conjunction $C$ located above the diagonal, excluding the values of the segments $\{0\} \times L_n$ and $L_n \times \{0\}$, since above them $C$ is equal to zero because it is an absorbing element. So, instead of seeing these values as a part of the discrete conjunction, let us see them as a simple triangular array, which satisfies certain monotone constraints between rows and columns. These triangular arrays are known as Gelfand-Tsetlin patterns, which were introduced in [69] by Gelfand and Tsetlin when studying the set of unimodular matrices.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 9**</caption><thead><tr><th>n</th><th>|A<sub>n</sub><sup>SMT</sup> ∩ A<sub>n</sub><sup>ABS</sup>|, |A<sub>n</sub><sup>SMT</sup> ∩ A<sub>n</sub><sup>NE</sup>|</th><th>C<sub>n</sub><sup>COM</sup> ∩ C<sub>n</sub><sup>IDE</sup></th><th>C<sub>n</sub><sup>IDE</sup></th></tr></thead><tbody><tr><td>1</td><td>2 = 2.10<sup>0</sup></td><td>1 = 1.10<sup>0</sup></td><td>1 = 1.10<sup>0</sup></td></tr><tr><td>2</td><td>5 = 5.10<sup>0</sup></td><td>2 = 2.10<sup>0</sup></td><td>4 = 4.10<sup>0</sup></td></tr><tr><td>3</td><td>18 = 1.8.10<sup>1</sup></td><td>8 = 8.10<sup>0</sup></td><td>64 = 6.4.10<sup>1</sup></td></tr><tr><td>4</td><td>102 = 1.02.10<sup>2</sup></td><td>64 = 6.4.10<sup>1</sup></td><td>4,096 = 4.096.10<sup>3</sup></td></tr><tr><td>5</td><td>970 = 9.7.10<sup>2</sup></td><td>1,024 = 1.024.10<sup>3</sup></td><td>≈ 1.0486.10<sup>6</sup></td></tr><tr><td>6</td><td>15,947 = 1.5947.10<sup>4</sup></td><td>32,768 = 3.2768.10<sup>4</sup></td><td>≈ 1.0737.10<sup>9</sup></td></tr><tr><td>7</td><td>≈ 4.5387.10<sup>5</sup></td><td>≈ 2.0972.10<sup>6</sup></td><td>≈ 4.398.10<sup>12</sup></td></tr><tr><td>8</td><td>≈ 2.2175.10<sup>7</sup></td><td>≈ 2.6844.10<sup>8</sup></td><td>≈ 7.2058.10<sup>16</sup></td></tr><tr><td>9</td><td>≈ 1.8464.10<sup>9</sup></td><td>≈ 6.8719.10<sup>10</sup></td><td>≈ 4.7224.10<sup>21</sup></td></tr><tr><td>10</td><td>≈ 2.6094.10<sup>11</sup></td><td>≈ 3.5184.10<sup>13</sup></td><td>≈ 1.2379.10<sup>27</sup></td></tr><tr><td>11</td><td>≈ 6.2454.10<sup>13</sup></td><td>≈ 3.6029.10<sup>16</sup></td><td>≈ 1.2981.10<sup>33</sup></td></tr><tr><td>12</td><td>≈ 2.5285.10<sup>16</sup></td><td>≈ 7.3787.10<sup>19</sup></td><td>≈ 5.4445.10<sup>39</sup></td></tr><tr><td>13</td><td>≈ 1.7304.10<sup>19</sup></td><td>≈ 3.0223.10<sup>23</sup></td><td>≈ 9.1344.10<sup>46</sup></td></tr></tbody></table>

Enumeration of the proposed formulas. From left to right: smooth discrete aggregation function with neutral element or with absorbing element, given by Proposition 5.2; commutative and idempotent discrete conjunctions, and idempotent discrete conjunctions, given by Proposition 5.4.

<figure><img src="image_2.png" /></figure>

Let us represent the set of idempotent discrete conjunctions as $C_n^{IDE}$. Looking at the above diagram, it is possible to decompose any idempotent discrete conjunction $C$ into two commutative idempotent discrete conjunctions. This arises from the fact that values of $C$ located above the diagonal $y = x$, with $x \in L_n$, and those below the diagonal, have no relation and are independent. Consequently, determining the enumeration of $|C_n^{COM} \cap C_n^{IDE}|$ provides a solution to the initial problem. We formalize this insight in the following proposition.

**Proposition 5.3.** Let $n \ge 1$, and let us denote by $C_n^{IDE}$ and $C_n^{COM}$ the sets of idempotent discrete conjunctions and commutative discrete conjunctions, respectively, defined on $L_n$. Let $\varphi : C_n^{IDE} \to (C_n^{COM} \cap C_n^{IDE}) \times (C_n^{COM} \cap C_n^{IDE})$ be defined by $\varphi(C) = (C_1, C_2)$, with $C \in C_n^{IDE}$, $C_1$ and $C_2$ are given point-wise by $C_1(x, y) = C_1(y, x) = C(x, y)$ and $C_2(x, y) = C_2(y, x) = C(y, x)$ for all $x \in L_n$ and $y \in [x, n]$. Then, the transformation $\varphi$ is a bijection.

*Proof.* First, $\varphi$ is well defined since $C_1$ and $C_2$ are commutative and idempotent by construction. Next, the injective and exhaustive nature of $\varphi$ follows directly from the expressions of $C_1$ and $C_2$, since the common points of $C_1$ and $C_2$ are only those on the diagonal of $C$. Then, it is a bijective transformation, and $\varphi^{-1}$ is given by $\varphi^{-1}(C_1, C_2) = C$, where $C(x, y) = C_1(x, y)$ and $C(y, x) = C_2(x, y)$ for all $x \in L_n$ and $y \in [x, n]$. $\square$

With this result, we can observe that $|C_n^{IDE}| = |C_n^{COM} \cap C_n^{IDE}|^2$ for all $n \ge 1$. To find the enumeration of $C_n^{COM} \cap C_n^{IDE}$, let us consider the values of any commutative and idempotent discrete conjunction $C$ located above the diagonal, excluding the values of the segments $\{0\} \times L_n$ and $L_n \times \{0\}$, since above them $C$ is equal to zero because it is an absorbing element. So, instead of seeing these values as a part of the discrete conjunction, let us see them as a simple triangular array, which satisfies certain monotone constraints between rows and columns. These triangular arrays are known as Gelfand-Tsetlin patterns, which were introduced in [69] by Gelfand and Tsetlin when studying the set of unimodular matrices.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Definition 5.2.** ([69, 70]) Let $n \ge 2$. A Gelfand-Tsetlin pattern $G = (g_{i,j})$ of size $n$ is a finite triangular array of natural numbers, with $1 \le i \le n$ and $1 \le j \le n - i + 1$, represented by

<figure><img src="image_2.png" /></figure>

which satisfies $g_{j+1,i} \le g_{j,i} \le g_{j,i+1}$, for all $1 \le i \le n-1$ and $1 \le j \le n-i$.

**Remark 5.2.** Since Gelfand-Tsetlin patterns appear in various fields of combinatorics, there are several ways to represent and index them. In this case, we have chosen to index them with the usual matrix indexing. See [69] or [70] for two examples where the indexing variables are different.

Our objective is to determine the enumeration of Gelfand-Tsetlin patterns that satisfy the condition $g_{n-i+1,i} = i$ for all $1 \le i \le n$. There are studies that have dealt with this problem. In [71], Zeilberger made an historical overview of enumerations for various structures, including plane partitions, alternating sign matrices and Young tableaux, among others. Zeilberger provided a formula for enumerating Gelfand-Tsetlin patterns meeting the condition of $g_{n-i+1,i} = i$ for all $1 \le i \le n$, though he did not provide a proof and categorized the process to obtain the formula as straightforward. Subsequently, Fischer, in [72, Theorem 1] established a formula for counting monotone triangles, where Gelfand-Tsetlin patterns' row monotonicity is replaced by strict monotonicity, and the condition to be met is $g_{n-i+1,i} = x_i$, where $\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{N}^n$ is a strictly increasing $n$-tuple. However, these structures are not relevant for modelling commutative idempotent discrete conjunctions. Years later, Fischer, in [73, Theorem 1], was the first to offer an enumeration for Gelfand-Tsetlin patterns with $g_{n-i+1,i} = x_i$, where $\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{N}^n$ is an increasing $n$-tuple. The result is as follows.

**Theorem 5.3.** ([72, Theorem 1]) Let $n \ge 2$, and let $\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{N}^n$ such that $x_i \le x_{i+1}$ for all $1 \le i \le n-1$. Then, the number of Gelfand-Tsetlin patterns $G = (g_{i,j})$ of size $n$ such that $g_{n-i+1,i} = x_i$ is given by

$$ \prod_{i=1}^{n} \prod_{j=i+1}^{n} \frac{x_j - x_i + j - i}{j - i}. \qquad (59) $$

Based on this result, we can establish the enumeration of $C_n^{\text{COM}} \cap C_n^{\text{IDE}}$.

**Proposition 5.4.** Let $n \ge 2$, and let us denote by $C_n^{\text{IDE}}$ and $C_n^{\text{COM}}$ the sets of idempotent discrete conjunctions and commutative discrete conjunctions, respectively, defined on $L_n$. Then, the following statements hold.

(i) The number of commutative and idempotent discrete conjunctions is given by

$$ |C_n^{\text{COM}} \cap C_n^{\text{IDE}}| = 2^{\frac{n(n-1)}{2}}. \qquad (60) $$

(ii) The number of idempotent discrete conjunctions is given by

$$ |C_n^{\text{IDE}}| = 2^{n(n-1)}. \qquad (61) $$

*Proof.* We prove each statement.

(i) Let $G_n$ be the set of Gelfand-Tsetlin patterns of size $n$ such that $g_{n-i+1,i} = i$, for all $1 \le i \le n$. First of all, the transformation $F: G_n \to C_n^{\text{COM}} \cap C_n^{\text{IDE}}$, defined as $F(G) = C$, where $C$ is given point-wise by $C(x, y) = 0$ if $x = 0$ or $y = 0$, or $C(x, y) = C(y, x) = g_{n-y+1,x}$ for all $1 \le x \le n$ and $1 \le y \le x-1$, is well
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Definition 5.2.** ([69, 70]) Let $n \ge 2$. A Gelfand-Tsetlin pattern $G = (g_{i,j})$ of size $n$ is a finite triangular array of natural numbers, with $1 \le i \le n$ and $1 \le j \le n - i + 1$, represented by

<table><tr><td>$g_{1,1}$</td><td>$g_{1,2}$</td><td>$g_{1,3}$</td><td>...</td><td>...</td><td>...</td><td>$g_{1,n}$</td></tr><tr><td>$g_{2,1}$</td><td>$g_{2,2}$</td><td>$g_{2,3}$</td><td>...</td><td>...</td><td>$g_{2,n-1}$</td><td></td></tr><tr><td>$g_{3,1}$</td><td>$g_{3,2}$</td><td>$g_{3,3}$</td><td>...</td><td>$g_{3,n-2}$</td><td></td><td></td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td></tr><tr><td>$g_{n-2,1}$</td><td>$g_{n-2,2}$</td><td>$g_{n-2,3}$</td><td></td><td></td><td></td><td></td></tr><tr><td>$g_{n-1,1}$</td><td>$g_{n-1,2}$</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>$g_{n,1}$</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>

which satisfies $g_{j+1,i} \le g_{j,i} \le g_{j,i+1}$, for all $1 \le i \le n-1$ and $1 \le j \le n-i$.

**Remark 5.2.** Since Gelfand-Tsetlin patterns appear in various fields of combinatorics, there are several ways to represent and index them. In this case, we have chosen to index them with the usual matrix indexing. See [69] or [70] for two examples where the indexing variables are different.

Our objective is to determine the enumeration of Gelfand-Tsetlin patterns that satisfy the condition $g_{n-i+1,i} = i$ for all $1 \le i \le n$. There are studies that have dealt with this problem. In [71], Zeilberger made an historical overview of enumerations for various structures, including plane partitions, alternating sign matrices and Young tableaux, among others. Zeilberger provided a formula for enumerating Gelfand-Tsetlin patterns meeting the condition of $g_{n-i+1,i} = i$ for all $1 \le i \le n$, though he did not provide a proof and categorized the process to obtain the formula as straightforward. Subsequently, Fischer, in [72, Theorem 1] established a formula for counting monotone triangles, where Gelfand-Tsetlin patterns' row monotonicity is replaced by strict monotonicity, and the condition to be met is $g_{n-i+1,i} = x_i$, where $\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{N}^n$ is a strictly increasing $n$-tuple. However, these structures are not relevant for modelling commutative idempotent discrete conjunctions. Years later, Fischer, in [73, Theorem 1], was the first to offer an enumeration for Gelfand-Tsetlin patterns with $g_{n-i+1,i} = x_i$, where $\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{N}^n$ is an increasing $n$-tuple. The result is as follows.

**Theorem 5.3.** ([72, Theorem 1]) Let $n \ge 2$, and let $\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{N}^n$ such that $x_i \le x_{i+1}$ for all $1 \le i \le n-1$. Then, the number of Gelfand-Tsetlin patterns $G = (g_{i,j})$ of size $n$ such that $g_{n-i+1,i} = x_i$ is given by

$$ \prod_{i=1}^{n} \prod_{j=i+1}^{n} \frac{x_j - x_i + j - i}{j - i}. \qquad (59) $$

Based on this result, we can establish the enumeration of $C_n^{\text{COM}} \cap C_n^{\text{IDE}}$.

**Proposition 5.4.** Let $n \ge 2$, and let us denote by $C_n^{\text{IDE}}$ and $C_n^{\text{COM}}$ the sets of idempotent discrete conjunctions and commutative discrete conjunctions, respectively, defined on $L_n$. Then, the following statements hold.

(i) The number of commutative and idempotent discrete conjunctions is given by

$$ |C_n^{\text{COM}} \cap C_n^{\text{IDE}}| = 2^{\frac{n(n-1)}{2}}. \qquad (60) $$

(ii) The number of idempotent discrete conjunctions is given by

$$ |C_n^{\text{IDE}}| = 2^{n(n-1)}. \qquad (61) $$

*Proof.* We prove each statement.

(i) Let $G_n$ be the set of Gelfand-Tsetlin patterns of size $n$ such that $g_{n-i+1,i} = i$, for all $1 \le i \le n$. First of all, the transformation $F: G_n \to C_n^{\text{COM}} \cap C_n^{\text{IDE}}$, defined as $F(G) = C$, where $C$ is given point-wise by $C(x, y) = 0$ if $x = 0$ or $y = 0$, or $C(x, y) = C(y, x) = g_{n-y+1,x}$ for all $1 \le x \le n$ and $1 \le y \le x-1$, is well
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

defined and also is a bijective correspondence. Then, applying Theorem 5.3 with $x = (1, 2, ..., n)$, we have

$$|C_n^{\text{COM}} \cap C_n^{\text{IDE}}| = \prod_{i=1}^{n} \prod_{j=i+1}^{n} \frac{j-i+j-i}{j-i} = \prod_{i=1}^{n} \prod_{j=i+1}^{n} 2 = 2^{\frac{n(n-1)}{2}},$$

as we wanted to prove.

(ii) It is deduced straightforwardly from statement (i) and the bijective transformation defined in Proposition 5.3.

<figure><img src="image_3.png" /></figure>

These two formulas are tabulated in Table 9. Considering the bijection outlined in Proposition 3.11, which relates discrete conjunctions and discrete disjunctions, and the bijection described in Proposition 4.1, linking discrete conjunctions and discrete implications, we can extend the two formulas proposed in Proposition 5.4 to the following cases:

* Statement (i) of Proposition 5.4 also applies for the enumeration of idempotent and commutative discrete disjunctions, and discrete implications satisfying $(CP(N_C))$ and $I(x, n - x) = n - x$, for all $x \in L_n$.

* Statement (ii) of Proposition 5.4 also applies for the enumeration of idempotent discrete disjunctions, and discrete implications satisfying $I(x, n - x) = n - x$, for all $x \in L_n$.

## 6. An overview of the existing enumerations and open issues

Since the advent of operators defined on $L_n$, there has been a notable surge in the interest surrounding their enumeration. Furthermore, these studies of enumeration have not only been confined to $L_n$ but have also emanated from diverse frameworks, including semigroups, plane partitions, and several classes of special matrices, among others. In this paper we have compiled and unified the established formulas for the main families of operators on $L_n$. In addition, during the process of reviewing the different formulas, we have corrected the proof of the enumeration of discrete uninorms having smooth and idempotent or smooth and Archimedean underlying operators in Proposition 3.20.

Moreover, we have made further progress in enumerating operators defined on $L_n$. By leveraging the existing characterizations of smooth discrete aggregation functions with neutral and absorbing elements, we have proposed their enumeration in Proposition 5.2, which is grounded in the cardinality of smooth discrete conjunctions and disjunctions. Then, we have also proposed an enumeration for commutative idempotent discrete conjunctions and idempotent discrete conjunctions in Proposition 5.4, based on Gelfand-Tsetlin patterns and their enumerations.

In Table 12, we provide a comprehensive summary of all known formulas, categorized by the family to which they belong to, the results with the formula, and the corresponding entry in the Online Encyclopedia of Integer Sequences (OEIS) [23]. Nonetheless, several enumerations of significant importance still pose an open problem. These are as follows:

* Smooth discrete aggregation functions. The study of this enumeration on $L_n$ has, to our knowledge, not yet been addressed. But it has been obtained computationally in OEIS/A306372 for matrices $A = (a_{i,j})$ of size $(n+1) \times (n+1)$, which satisfy $a_{1,1} = 1$, $a_{n+1,n+1} = n+1$, the entries in each row and column are increasing, $a_{i+1,j} - a_{i,j} \le 1$ and $a_{j,i+1} - a_{j,i} \le 1$ for all $i \in \{1, ..., n-1\}$ and $j \in \{1, ..., n\}$, whose enumeration clearly coincides with the number of smooth discrete aggregation functions. This enumeration is shown in Table 10.

* Smooth and commutative discrete aggregation functions. Its enumeration has not been addressed on $L_n$ nor tabulated in OEIS. The authors of this paper, using computational methods, have established their enumeration for $n \le 6$, which is shown in Table 10, and tabulated in OEIS/A367432.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

defined and also is a bijective correspondence. Then, applying Theorem 5.3 with $x = (1, 2, ..., n)$, we have

$$|C_n^{\text{COM}} \cap C_n^{\text{IDE}}| = \prod_{i=1}^{n} \prod_{j=i+1}^{n} \frac{j-i+j-i}{j-i} = \prod_{i=1}^{n} \prod_{j=i+1}^{n} 2 = 2^{\frac{n(n-1)}{2}},$$

as we wanted to prove.

(ii) It is deduced straightforwardly from statement (i) and the bijective transformation defined in Proposition 5.3. □

These two formulas are tabulated in Table 9. Considering the bijection outlined in Proposition 3.11, which relates discrete conjunctions and discrete disjunctions, and the bijection described in Proposition 4.1, linking discrete conjunctions and discrete implications, we can extend the two formulas proposed in Proposition 5.4 to the following cases:

* Statement (i) of Proposition 5.4 also applies for the enumeration of idempotent and commutative discrete disjunctions, and discrete implications satisfying $(CP(N_C))$ and $I(x, n - x) = n - x$, for all $x \in L_n$.

* Statement (ii) of Proposition 5.4 also applies for the enumeration of idempotent discrete disjunctions, and discrete implications satisfying $I(x, n - x) = n - x$, for all $x \in L_n$.

## 6. An overview of the existing enumerations and open issues

Since the advent of operators defined on $L_n$, there has been a notable surge in the interest surrounding their enumeration. Furthermore, these studies of enumeration have not only been confined to $L_n$ but have also emanated from diverse frameworks, including semigroups, plane partitions, and several classes of special matrices, among others. In this paper we have compiled and unified the established formulas for the main families of operators on $L_n$. In addition, during the process of reviewing the different formulas, we have corrected the proof of the enumeration of discrete uninorms having smooth and idempotent or smooth and Archimedean underlying operators in Proposition 3.20.

Moreover, we have made further progress in enumerating operators defined on $L_n$. By leveraging the existing characterizations of smooth discrete aggregation functions with neutral and absorbing elements, we have proposed their enumeration in Proposition 5.2, which is grounded in the cardinality of smooth discrete conjunctions and disjunctions. Then, we have also proposed an enumeration for commutative idempotent discrete conjunctions and idempotent discrete conjunctions in Proposition 5.4, based on Gelfand-Tsetlin patterns and their enumerations.

In Table 12, we provide a comprehensive summary of all known formulas, categorized by the family to which they belong to, the results with the formula, and the corresponding entry in the Online Encyclopedia of Integer Sequences (OEIS) [23]. Nonetheless, several enumerations of significant importance still pose an open problem. These are as follows:

* Smooth discrete aggregation functions. The study of this enumeration on $L_n$ has, to our knowledge, not yet been addressed. But it has been obtained computationally in OEIS/A306372 for matrices $A = (a_{i,j})$ of size $(n+1) \times (n+1)$, which satisfy $a_{1,1} = 1$, $a_{n+1,n+1} = n+1$, the entries in each row and column are increasing, $a_{i+1,j} - a_{i,j} \le 1$ and $a_{j,i+1} - a_{j,i} \le 1$ for all $i \in \{1, ..., n-1\}$ and $j \in \{1, ..., n\}$, whose enumeration clearly coincides with the number of smooth discrete aggregation functions. This enumeration is shown in Table 10.

* Smooth and commutative discrete aggregation functions. Its enumeration has not been addressed on $L_n$ nor tabulated in OEIS. The authors of this paper, using computational methods, have established their enumeration for $n \le 6$, which is shown in Table 10, and tabulated in OEIS/A367432.
* Smooth and commutative discrete conjunctions. Theorem 3.1 establishes a bijection between smooth discrete conjunctions and alternating sign matrices. Munar et al. in [20, Proposition 3.6] already observed that the same correspondence holds between smooth and commutative discrete conjunctions and symmetric alternating sign matrices. However, Bousquet-Mélou and Habsieger in [74] indicated that a formula for the enumeration of this particular type of matrices remains an open problem, providing their enumeration for $n \le 20$. Recently, Behrend et al. in [75] reaffirmed that this remains unsolved. Some values of this sequence are exemplified in Table 10. It is tabulated in OEIS/A005163.

* Discrete t-norms. De Baets and Mesiar in [5] determined its cardinality for $n \le 13$ and, since then, no further progress has been known. In part, the difficulty in determining a formula is due to the associativity property. It should be noted that the formula for the number of commutative conjunctions with neutral element $n$ given in Proposition 3.9 serves as an upper bound on the number of discrete t-norms. The known values are illustrated in Table 10 and it is tabulated in OEIS/A030453.

* Archimedean discrete t-norms. To our knowledge, no progress has been made on this enumeration. We can only highlight the work of Petrik and Vetterlein in [76], where they propose an algorithm to generate the enumeration of these operators, although they only manage to reach $n \le 12$. These values are listed in Table 10. It is tabulated in OEIS/A253948.

* Discrete uninorms. Its structure as a combination of a discrete t-norm and a discrete t-conorm suggests that its enumeration is not straightforward. Indeed, no study has been found in the literature that deals with their enumeration. We have established their enumeration for $n \le 9$ using computational methods, as shown in Table 10.

* Discrete implications satisfying (EP), $(CP(N))$, $(LI(T))$ and $(MP(T))$ in general. Nachtegael and Kerre in [6] computationally enumerated discrete implications satisfying (EP) for $n \le 5$ and, to the best of our knowledge, no further study has dealt with the enumeration of the other properties. For $(CP(N))$, Munar et al. in [19] addressed it when $N = N_C$, but the general case is still unexplored. The authors of this paper, using computational methods, have established their enumeration for $n \le 5$, which are shown in Table 11 and tabulated it in OEIS/A367444, OEIS/A367445, OEIS/A367447 and OEIS/A367446, respectively. It is important to note that the obtained enumeration for $(CP(N))$ encompasses all those discrete implications satisfying $(CP(N))$ with some negation $N$, i.e., the negation $N$ is not fixed and if one discrete implication satisfies the contrapositive symmetry with more than one discrete negation, it only appears once in the enumeration. The enumerations of $(LI(T))$ and $(MP(T))$ are computed in an analogous way. Observe, finally, that the number of discrete implications satisfying $(MT(T,N))$ for some discrete t-conorm $T$ and when the discrete negation is $N_C$ is also shown in Table 11. It coincides with the number of discrete implications satisfying $(MP(T))$ with respect to some discrete t-norm $T$, since in this case the bijection of Proposition 4.1 transforms one class into the other.

To conclude, the examination of operators defined on $L_n$ has a pronounced computational dimension. In light of this, it is worth noting that some of the authors of this study developed DiscreteFuzzyOperators in [77], a pure-Python package devoted to managing some classes of operators mentioned in this paper, along with the incorporation of most of the formulas for their enumeration compiled herein. Additionally, we have taken the initiative to construct the Discrete Dataset [78], wherein, when feasible, we have generated all operators from diverse classes, for some values of $n$. It is publicly accessible in [78], with the aim of facilitating future research work. The repository also includes the code used to generate the different enumerations that have appeared in the paper.

## Acknowledgments

This work was partially supported by the R+D+i Project PID2020-113870GB-I00-"Desarrollo de herramientas de Soft Computing para la Ayuda al Diagnóstico Clínico y a la Gestión de Emergencias (HESOCODICE)", funded by MCIN/AEI/10.13039/501100011033/.
* Smooth and commutative discrete conjunctions. Theorem 3.1 establishes a bijection between smooth discrete conjunctions and alternating sign matrices. Munar et al. in [20, Proposition 3.6] already observed that the same correspondence holds between smooth and commutative discrete conjunctions and symmetric alternating sign matrices. However, Bousquet-Mélou and Habsieger in [74] indicated that a formula for the enumeration of this particular type of matrices remains an open problem, providing their enumeration for $n \le 20$. Recently, Behrend et al. in [75] reaffirmed that this remains unsolved. Some values of this sequence are exemplified in Table 10. It is tabulated in OEIS/A005163.

* Discrete t-norms. De Baets and Mesiar in [5] determined its cardinality for $n \le 13$ and, since then, no further progress has been known. In part, the difficulty in determining a formula is due to the associativity property. It should be noted that the formula for the number of commutative conjunctions with neutral element $n$ given in Proposition 3.9 serves as an upper bound on the number of discrete t-norms. The known values are illustrated in Table 10 and it is tabulated in OEIS/A030453.

* Archimedean discrete t-norms. To our knowledge, no progress has been made on this enumeration. We can only highlight the work of Petrik and Vetterlein in [76], where they propose an algorithm to generate the enumeration of these operators, although they only manage to reach $n \le 12$. These values are listed in Table 10. It is tabulated in OEIS/A253948.

* Discrete uninorms. Its structure as a combination of a discrete t-norm and a discrete t-conorm suggests that its enumeration is not straightforward. Indeed, no study has been found in the literature that deals with their enumeration. We have established their enumeration for $n \le 9$ using computational methods, as shown in Table 10.

* Discrete implications satisfying (EP), $(CP(N))$, $(LI(T))$ and $(MP(T))$ in general. Nachtegael and Kerre in [6] computationally enumerated discrete implications satisfying (EP) for $n \le 5$ and, to the best of our knowledge, no further study has dealt with the enumeration of the other properties. For $(CP(N))$, Munar et al. in [19] addressed it when $N = N_C$, but the general case is still unexplored. The authors of this paper, using computational methods, have established their enumeration for $n \le 5$, which are shown in Table 11 and tabulated it in OEIS/A367444, OEIS/A367445, OEIS/A367447 and OEIS/A367446, respectively. It is important to note that the obtained enumeration for $(CP(N))$ encompasses all those discrete implications satisfying $(CP(N))$ with some negation $N$, i.e., the negation $N$ is not fixed and if one discrete implication satisfies the contrapositive symmetry with more than one discrete negation, it only appears once in the enumeration. The enumerations of $(LI(T))$ and $(MP(T))$ are computed in an analogous way. Observe, finally, that the number of discrete implications satisfying $(MT(T,N))$ for some discrete t-conorm $T$ and when the discrete negation is $N_C$ is also shown in Table 11. It coincides with the number of discrete implications satisfying $(MP(T))$ with respect to some discrete t-norm $T$, since in this case the bijection of Proposition 4.1 transforms one class into the other.

To conclude, the examination of operators defined on $L_n$ has a pronounced computational dimension. In light of this, it is worth noting that some of the authors of this study developed DiscreteFuzzyOperators in [77], a pure-Python package devoted to managing some classes of operators mentioned in this paper, along with the incorporation of most of the formulas for their enumeration compiled herein. Additionally, we have taken the initiative to construct the Discrete Dataset [78], wherein, when feasible, we have generated all operators from diverse classes, for some values of $n$. It is publicly accessible in [78], with the aim of facilitating future research work. The repository also includes the code used to generate the different enumerations that have appeared in the paper.

## Acknowledgments

This work was partially supported by the R+D+i Project PID2020-113870GB-I00-"Desarrollo de herramientas de Soft Computing para la Ayuda al Diagnóstico Clínico y a la Gestión de Emergencias (HESOCODICE)", funded by MCIN/AEI/10.13039/501100011033/.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 10**</caption><thead><tr><th>n</th><th>A<sup>SMT</sup><sub>n</sub></th><th>A<sup>SMT</sup><sub>n</sub> &cap; A<sup>COM</sup><sub>n</sub></th><th>C<sup>SMT</sup><sub>n</sub> &cap; C<sup>COM</sup><sub>n</sub></th><th>T<sub>n</sub></th><th>T<sup>ARC</sup><sub>n</sub></th><th>U<sub>n</sub></th></tr></thead><tbody><tr><td>1</td><td>4 = 4 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>44 = 4.4 &middot; 10<sup>1</sup></td><td>10 = 1 &middot; 10<sup>1</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>1,236 = 1.236 &middot; 10<sup>3</sup></td><td>80 = 8 &middot; 10<sup>1</sup></td><td>5 = 5 &middot; 10<sup>0</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>22 = 2.2 &middot; 10<sup>1</sup></td></tr><tr><td>4</td><td>84,910 = 8.491 &middot; 10<sup>3</sup></td><td>1,008 = 1.008 &middot; 10<sup>3</sup></td><td>16 = 1.6 &middot; 10<sup>1</sup></td><td>22 = 2.2 &middot; 10<sup>1</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td><td>92 = 9.2 &middot; 10<sup>1</sup></td></tr><tr><td>5</td><td>&asymp; 1.4024 &middot; 10<sup>7</sup></td><td>19,764 = 1.9764 &middot; 10<sup>4</sup></td><td>67 = 6.7 &middot; 10<sup>1</sup></td><td>94 = 9.4 &middot; 10<sup>1</sup></td><td>22 = 2.2 &middot; 10<sup>1</sup></td><td>426 = 4.26 &middot; 10<sup>2</sup></td></tr><tr><td>6</td><td>&asymp; 5.531 &middot; 10<sup>9</sup></td><td>600,028 &asymp; 6.0003 &middot; 10<sup>5</sup></td><td>368 = 3.68 &middot; 10<sup>2</sup></td><td>451 = 4.51 &middot; 10<sup>2</sup></td><td>95 = 9.5 &middot; 10<sup>1</sup></td><td>2,146 = 2.146 &middot; 10<sup>3</sup></td></tr><tr><td>7</td><td>&asymp; 5.1922 &middot; 10<sup>12</sup></td><td>—</td><td>2,630 = 2.63 &middot; 10<sup>3</sup></td><td>2,386 = 2.386 &middot; 10<sup>3</sup></td><td>471 = 4.71 &middot; 10<sup>2</sup></td><td>11,634 = 1.1634 &middot; 10<sup>4</sup></td></tr><tr><td>8</td><td>&asymp; 1.1583 &middot; 10<sup>16</sup></td><td>—</td><td>24,376 = 2.4376 &middot; 10<sup>4</sup></td><td>13,775 = 1.3775 &middot; 10<sup>4</sup></td><td>2,670 = 2.67 &middot; 10<sup>3</sup></td><td>67,472 = 6.7472 &middot; 10<sup>4</sup></td></tr><tr><td>9</td><td>&asymp; 6.1353 &middot; 10<sup>19</sup></td><td>—</td><td>293,770 = 2.9377 &middot; 10<sup>5</sup></td><td>86,417 = 8.6417 &middot; 10<sup>4</sup></td><td>17,387 = 1.7387 &middot; 10<sup>4</sup></td><td>417,650 = 4.1765 &middot; 10<sup>5</sup></td></tr><tr><td>10</td><td>&asymp; 7.7117 &middot; 10<sup>25</sup></td><td>—</td><td>&asymp; 4.6106 &middot; 10<sup>6</sup></td><td>&asymp; 5.9049 &middot; 10<sup>5</sup></td><td>&asymp; 1.3175 &middot; 10<sup>5</sup></td><td>—</td></tr><tr><td>11</td><td>&asymp; 2.2994 &middot; 10<sup>28</sup></td><td>—</td><td>&asymp; 9.4081 &middot; 10<sup>7</sup></td><td>&asymp; 4.446 &middot; 10<sup>6</sup></td><td>&asymp; 1.1841 &middot; 10<sup>6</sup></td><td>—</td></tr><tr><td>12</td><td>&asymp; 1.626 &middot; 10<sup>33</sup></td><td>—</td><td>&asymp; 2.4927 &middot; 10<sup>9</sup></td><td>&asymp; 3.7869 &middot; 10<sup>7</sup></td><td>&asymp; 1.2897 &middot; 10<sup>7</sup></td><td>—</td></tr><tr><td>13</td><td>&asymp; 2.7265 &middot; 10<sup>38</sup></td><td>—</td><td>&asymp; 8.5828 &middot; 10<sup>10</sup></td><td>&asymp; 3.8255 &middot; 10<sup>8</sup></td><td>—</td><td>—</td></tr><tr><td></td><td>OEIS/A306372</td><td>OEIS/A367432</td><td>OEIS/A005163</td><td>OEIS/A030453</td><td>OEIS/A253948</td><td>OEIS/A367442</td></tr></tbody></table>

Enumeration of some classes of discrete aggregation functions, for $n \le 13$, obtained using computational methods since a general formula for their enumeration has not been found. From left to right: smooth discrete aggregation functions, smooth and commutative discrete aggregation functions, smooth and commutative discrete conjunctions, discrete t-norms, Archimedean discrete t-norms and discrete uninorms.

<table><caption>**Table 11**</caption><thead><tr><th>n</th><th>I<sup>EP</sup><sub>n</sub></th><th>I<sup>CP</sup><sub>n</sub></th><th>I<sup>MP</sup><sub>n</sub></th><th>I<sup>LI</sup><sub>n</sub></th><th>I<sup>MT(NC)</sup><sub>n</sub></th></tr></thead><tbody><tr><td>1</td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>10 = 1 &middot; 10<sup>1</sup></td><td>8 = 8 &middot; 10<sup>0</sup></td><td>9 = 9 &middot; 10<sup>0</sup></td><td>9 = 9 &middot; 10<sup>0</sup></td><td>9 = 9 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>165 = 1.65 &middot; 10<sup>2</sup></td><td>99 = 9.9 &middot; 10<sup>1</sup></td><td>519 = 7.7 &middot; 10<sup>1</sup></td><td>102 = 1.02 &middot; 10<sup>2</sup></td><td>519 = 7.7 &middot; 10<sup>1</sup></td></tr><tr><td>4</td><td>3.863 = 3.863 &middot; 10<sup>3</sup></td><td>2,828 = 2.828 &middot; 10<sup>3</sup></td><td>150,120 &asymp; 1.5012 &middot; 10<sup>3</sup></td><td>1,529 = 1.529 &middot; 10<sup>3</sup></td><td>150,120 &asymp; 1.5012 &middot; 10<sup>3</sup></td></tr><tr><td>5</td><td>117,096 &asymp; 1.171 &middot; 10<sup>5</sup></td><td>152,474 &asymp; 1.5247 &middot; 10<sup>5</sup></td><td>202,728,377 &asymp; 2.0273 &middot; 10<sup>8</sup></td><td>28,702 = 2.8702 &middot; 10<sup>4</sup></td><td>202,728,377 &asymp; 2.0273 &middot; 10<sup>8</sup></td></tr><tr><td></td><td>OEIS/A367444</td><td>OEIS/A367445</td><td>OEIS/A367446</td><td>OEIS/A367447</td><td>OEIS/A367446</td></tr></tbody></table>

Enumeration of some classes of discrete implications, for $n \le 5$, obtained using computational methods since a general formula for their enumeration has not been found. From left to right: discrete implications satisfying (EP), (CP(N)) with respect to some discrete negation N, (MP(T)) with respect to some discrete t-norm T, (LI(T)) with respect to some discrete t-norm T, and (MT(T, N)) with respect to some discrete t-norm T and the classical discrete negation N<sub>C</sub>.

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A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 10**</caption><thead><tr><th>n</th><th>|A<sup>SMT</sup><sub>n</sub>|</th><th>|A<sup>SMT</sup><sub>n</sub> &cap; A<sup>COM</sup><sub>n</sub>|</th><th>|C<sup>SMT</sup><sub>n</sub> &cap; C<sup>COM</sup><sub>n</sub>|</th><th>|T<sub>n</sub>|</th><th>|T<sup>ARC</sup><sub>n</sub>|</th><th>|U<sup>n</sup>|</th></tr></thead><tbody><tr><td>1</td><td>4 = 4 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>44 = 4.4 &middot; 10<sup>1</sup></td><td>10 = 1 &middot; 10<sup>1</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>1,236 = 1.236 &middot; 10<sup>3</sup></td><td>80 = 8 &middot; 10<sup>1</sup></td><td>5 = 5 &middot; 10<sup>0</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>22 = 2.2 &middot; 10<sup>1</sup></td></tr><tr><td>4</td><td>84,910 = 8.491 &middot; 10<sup>3</sup></td><td>1,008 = 1.008 &middot; 10<sup>3</sup></td><td>16 = 1.6 &middot; 10<sup>1</sup></td><td>22 = 2.2 &middot; 10<sup>1</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td><td>92 = 9.2 &middot; 10<sup>1</sup></td></tr><tr><td>5</td><td>≈ 1.4024 &middot; 10<sup>7</sup></td><td>19,764 = 1.9764 &middot; 10<sup>4</sup></td><td>67 = 6.7 &middot; 10<sup>1</sup></td><td>94 = 9.4 &middot; 10<sup>1</sup></td><td>22 = 2.2 &middot; 10<sup>1</sup></td><td>426 = 4.26 &middot; 10<sup>2</sup></td></tr><tr><td>6</td><td>≈ 5.531 &middot; 10<sup>9</sup></td><td>600,028 ≈ 6.0003 &middot; 10<sup>5</sup></td><td>368 = 3.68 &middot; 10<sup>2</sup></td><td>451 = 4.51 &middot; 10<sup>2</sup></td><td>95 = 9.5 &middot; 10<sup>1</sup></td><td>2,146 = 2.146 &middot; 10<sup>3</sup></td></tr><tr><td>7</td><td>≈ 5.1922 &middot; 10<sup>12</sup></td><td>—</td><td>2,630 = 2.63 &middot; 10<sup>3</sup></td><td>2,386 = 2.386 &middot; 10<sup>3</sup></td><td>471 = 4.71 &middot; 10<sup>2</sup></td><td>11,634 = 1.1634 &middot; 10<sup>4</sup></td></tr><tr><td>8</td><td>≈ 1.1583 &middot; 10<sup>16</sup></td><td>—</td><td>24,376 = 2.4376 &middot; 10<sup>4</sup></td><td>13,775 = 1.3775 &middot; 10<sup>4</sup></td><td>2,670 = 2.67 &middot; 10<sup>3</sup></td><td>67,472 = 6.7472 &middot; 10<sup>4</sup></td></tr><tr><td>9</td><td>≈ 6.1353 &middot; 10<sup>19</sup></td><td>—</td><td>293,770 = 2.9377 &middot; 10<sup>5</sup></td><td>86,417 = 8.6417 &middot; 10<sup>4</sup></td><td>17,387 = 1.7387 &middot; 10<sup>4</sup></td><td>417,650 = 4.1765 &middot; 10<sup>5</sup></td></tr><tr><td>10</td><td>≈ 7.7117 &middot; 10<sup>25</sup></td><td>—</td><td>≈ 4.6106 &middot; 10<sup>6</sup></td><td>≈ 5.9049 &middot; 10<sup>5</sup></td><td>≈ 1.3175 &middot; 10<sup>5</sup></td><td>—</td></tr><tr><td>11</td><td>≈ 2.2994 &middot; 10<sup>28</sup></td><td>—</td><td>≈ 9,4081 &middot; 10<sup>7</sup></td><td>≈ 4.446 &middot; 10<sup>6</sup></td><td>≈ 1.1841 &middot; 10<sup>6</sup></td><td>—</td></tr><tr><td>12</td><td>≈ 1.626 &middot; 10<sup>33</sup></td><td>—</td><td>≈ 2.4927 &middot; 10<sup>9</sup></td><td>≈ 3.7869 &middot; 10<sup>7</sup></td><td>≈ 1.2897 &middot; 10<sup>7</sup></td><td>—</td></tr><tr><td>13</td><td>≈ 2.7265 &middot; 10<sup>38</sup></td><td>—</td><td>≈ 8.5828 &middot; 10<sup>10</sup></td><td>≈ 3.8255 &middot; 10<sup>8</sup></td><td>—</td><td>—</td></tr><tr><td></td><td>OEIS/A306372</td><td>OEIS/A367432</td><td>OEIS/A005163</td><td>OEIS/A030453</td><td>OEIS/A253948</td><td>OEIS/A367442</td></tr></tbody></table>

Enumeration of some classes of discrete aggregation functions, for $n \le 13$, obtained using computational methods since a general formula for their enumeration has not been found. From left to right: smooth discrete aggregation functions, smooth and commutative discrete aggregation functions, smooth and commutative discrete conjunctions, discrete t-norms, Archimedean discrete t-norms and discrete uninorms.

<table><caption>**Table 11**</caption><thead><tr><th>n</th><th>|I<sup>EP</sup><sub>n</sub>|</th><th>|I<sup>CP</sup><sub>n</sub>|</th><th>|I<sup>MP</sup><sub>n</sub>|</th><th>|I<sup>LI</sup><sub>n</sub>|</th><th>|I<sup>MT(NC)</sup><sub>n</sub>|</th></tr></thead><tbody><tr><td>1</td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>10 = 1 &middot; 10<sup>1</sup></td><td>8 = 8 &middot; 10<sup>0</sup></td><td>9 = 9 &middot; 10<sup>0</sup></td><td>9 = 9 &middot; 10<sup>0</sup></td><td>9 = 9 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>165 = 1.65 &middot; 10<sup>2</sup></td><td>99 = 9.9 &middot; 10<sup>1</sup></td><td>519 = 7.7 &middot; 10<sup>1</sup></td><td>102 = 1.02 &middot; 10<sup>2</sup></td><td>519 = 7.7 &middot; 10<sup>1</sup></td></tr><tr><td>4</td><td>3.863 = 3.863 &middot; 10<sup>3</sup></td><td>2,828 = 2.828 &middot; 10<sup>3</sup></td><td>150,120 ≈ 1.5012 &middot; 10<sup>3</sup></td><td>1,529 = 1.529 &middot; 10<sup>3</sup></td><td>150,120 ≈ 1.5012 &middot; 10<sup>3</sup></td></tr><tr><td>5</td><td>117,096 ≈ 1.171 &middot; 10<sup>5</sup></td><td>152,474 ≈ 1.5247 &middot; 10<sup>5</sup></td><td>202,728,377 ≈ 2.0273 &middot; 10<sup>8</sup></td><td>28,702 = 2.8702 &middot; 10<sup>4</sup></td><td>202,728,377 ≈ 2.0273 &middot; 10<sup>8</sup></td></tr><tr><td></td><td>OEIS/A367444</td><td>OEIS/A367445</td><td>OEIS/A367446</td><td>OEIS/A367447</td><td>OEIS/A367446</td></tr></tbody></table>

Enumeration of some classes of discrete implications, for $n \le 5$, obtained using computational methods since a general formula for their enumeration has not been found. From left to right: discrete implications satisfying (EP), $(CP(N))$ with respect to some discrete negation $N$, $(MP(T))$ with respect to some discrete t-norm $T$, $(LI(T))$ with respect to some discrete t-norm $T$, and $(MT(T, N))$ with respect to some discrete t-norm $T$ and the classical discrete negation $N_C$.

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[63] M. Baczyński, B. Jayaram, S. Massanet, and J. Torrens, "Fuzzy implications: Past, present, and future," in *Springer Handbook of Computational Intelligence* (J. Kacprzyk and W. Pedrycz, eds.), pp. 183–202, Springer, 2015.

[64] M. Mas, M. Monserrat, and J. Torrens, "S-implications and R-implications on a finite chain," *Kybernetika*, vol. 40, pp. 3–20, 2004.

[65] C. Krattenthaler, "A determinant evaluation and some enumeration results for plane partitions," in *Number-Theoretic Analysis* (E. Hlawka and R. Tichy, eds.), pp. 121–131, Springer, 1990.

[66] C. Krattenthaler, "Generating functions for plane partitions of a given shape," *Manuscripta mathematica*, vol. 69, pp. 173–201, 1990.

[67] B. Lindström, "On the vector representations of induced matroids," *Bulletin of the London Mathematical Society*, vol. 5, pp. 85–90, 1973.

[68] I. Gessel and G. Viennot, "Binomial determinants, paths, and hook length formulae," *Advances in Mathematics*, vol. 58, pp. 300–321, 1985.

[69] I. Gelfand and M. Tsetlin, "Finite-dimensional representations of the group of unimodular matrices," *Proceedings of the USSR Academy of Sciences*, vol. 71, pp. 825–828, 1950.

[70] T. Tokuyama, "A generating function of strict gelfand patterns and some formulas on characters of general linear groups," *Journal of The Mathematical Society of Japan*, vol. 40, pp. 671–685, 1988.

[71] D. Zeilberger, "Dave robbins' art of guessing," *Advances in Applied Mathematics*, vol. 34, pp. 939–954, 2005.

[72] I. Fischer, "The number of monotone triangles with prescribed bottom row," *Advances in Applied Mathematics*, vol. 37, pp. 249–267, 2006.

[73] I. Fischer, "Sequences of labeled trees related to gelfand–tsetlin patterns," *Advances in Applied Mathematics*, vol. 49, pp. 165–195, 2012.

[74] M. Bousquet-Mélou and L. Habsieger, "Sur les matrices à signes alternants," *Discrete Mathematics*, vol. 139, pp. 57–72, 1995.

[75] R. Behrend, I. Fischer, and M. Konvalinka, "Diagonally and antidiagonally symmetric alternating sign matrices of odd order," *Advances in Mathematics*, vol. 315, pp. 324–365, 2017.

[76] M. Petrik and T. Vetterlein, "Algorithm to generate the Archimedean, finite, negative tomonoids," in *2014 Joint 7th International Conference on Soft Computing and Intelligent Systems (SCIS) and 15th International Symposium on Advanced Intelligent Systems (ISIS)*, pp. 42–47, 2014.

[77] M. Munar, S. Massanet, and D. Ruiz-Aguilera, "Discrete Fuzzy Operators - A Python library for computing with fuzzy operators (version 1.13)," 2022. URL: https://doi.org/10.5281/zenodo.6503674.

[78] M. Munar, M. Couceiro, S. Massanet, and D. Ruiz-Aguilera, "Discrete Dataset - A dataset for computing with operators defined on a finite chain (version 1.0)," 2023. URL: https://doi.org/10.5281/zenodo.10184481.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with
new results

[53] F. Zhang, *Matrix Theory: Basic Results and Techniques*. Springer, 2011.

[54] C. Bejines, M. Bruteničová, M. Chasco, J. Elorza, and V. Janiš, "The number of t-norms on some special lattices," *Fuzzy Sets and Systems*, vol. 408, pp. 26–43, 2021.

[55] C. Bejines, "T-norms and t-conorms on a family of lattices," *Fuzzy Sets and Systems*, vol. 439, pp. 55–74, 2022.

[56] G. Mayor and J. Torrens, "On a class of operators for expert systems," *International Journal of Intelligent Systems*, vol. 8, pp. 771–778, 1993.

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[59] J. V. Riera and J. Torrens, "Uninorms and nullnorms on the set of discrete fuzzy numbers," in *Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology*, pp. 59–66, 2011.

[60] J. Fodor and M. Roubens, *Fuzzy Preference Modelling and Multicriteria Decision Support*. Springer, 1 ed., 1994.

[61] M. Baczyński and B. Jayaram, *Fuzzy Implications*. Studies in Fuzziness and Soft Computing, Springer, 2008.

[62] M. Mas, M. Monserrat, J. Torrens, and E. Trillas, "A survey on fuzzy implication functions," *IEEE Transactions on Fuzzy Systems*, vol. 15, pp. 1107–1121, 2007.

[63] M. Baczyński, B. Jayaram, S. Massanet, and J. Torrens, "Fuzzy implications: Past, present, and future," in *Springer Handbook of Computational Intelligence* (J. Kacprzyk and W. Pedrycz, eds.), pp. 183–202, Springer, 2015.

[64] M. Mas, M. Monserrat, and J. Torrens, "S-implications and R-implications on a finite chain," *Kybernetika*, vol. 40, pp. 3–20, 2004.

[65] C. Krattenthaler, "A determinant evaluation and some enumeration results for plane partitions," in *Number-Theoretic Analysis* (E. Hlawka and R. Tichy, eds.), pp. 121–131, Springer, 1990.

[66] C. Krattenthaler, "Generating functions for plane partitions of a given shape," *Manuscripta mathematica*, vol. 69, pp. 173–201, 1990.

[67] B. Lindström, "On the vector representations of induced matroids," *Bulletin of the London Mathematical Society*, vol. 5, pp. 85–90, 1973.

[68] I. Gessel and G. Viennot, "Binomial determinants, paths, and hook length formulae," *Advances in Mathematics*, vol. 58, pp. 300–321, 1985.

[69] I. Gelfand and M. Tsetlin, "Finite-dimensional representations of the group of unimodular matrices," *Proceedings of the USSR Academy of Sciences*, vol. 71, pp. 825–828, 1950.

[70] T. Tokuyama, "A generating function of strict gelfand patterns and some formulas on characters of general linear groups," *Journal of The Mathematical Society of Japan*, vol. 40, pp. 671–685, 1988.

[71] D. Zeilberger, "Dave robbins' art of guessing," *Advances in Applied Mathematics*, vol. 34, pp. 939–954, 2005.

[72] I. Fischer, "The number of monotone triangles with prescribed bottom row," *Advances in Applied Mathematics*, vol. 37, pp. 249–267, 2006.

[73] I. Fischer, "Sequences of labeled trees related to gelfand–tsetlin patterns," *Advances in Applied Mathematics*, vol. 49, pp. 165–195, 2012.

[74] M. Bousquet-Mélou and L. Habsieger, "Sur les matrices à signes alternants," *Discrete Mathematics*, vol. 139, pp. 57–72, 1995.

[75] R. Behrend, I. Fischer, and M. Konvalinka, "Diagonally and antidiagonally symmetric alternating sign matrices of odd order," *Advances in Mathematics*, vol. 315, pp. 324–365, 2017.

[76] M. Petrik and T. Vetterlein, "Algorithm to generate the Archimedean, finite, negative tomonoids," in *2014 Joint 7th International Conference on Soft Computing and Intelligent Systems (SCIS) and 15th International Symposium on Advanced Intelligent Systems (ISIS)*, pp. 42–47, 2014.

[77] M. Munar, S. Massanet, and D. Ruiz-Aguilera, "Discrete Fuzzy Operators - A Python library for computing with fuzzy operators (version 1.13)," 2022. URL: https://doi.org/10.5281/zenodo.6503674.

[78] M. Munar, M. Couceiro, S. Massanet, and D. Ruiz-Aguilera, "Discrete Dataset - A dataset for computing with operators defined on a finite chain (version 1.0)," 2023. URL: https://doi.org/10.5281/zenodo.10184481.
<table><caption>A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results</caption><thead><tr><th>Class</th><th>Subclass</th><th>Result</th><th>Original source</th><th>OEIS</th></tr></thead><tbody><tr><td rowspan="2">Discrete negations</td><td>Discrete negations</td><td>Prop. 2.1</td><td>[19]</td><td>A001700</td></tr><tr><td>Over anti-diagonal discrete negations</td><td>Prop. 2.2</td><td>[8]</td><td>A000108</td></tr><tr><td rowspan="15">Discrete aggregation functions</td><td>Discrete aggregation functions</td><td>Prop. 3.1</td><td>[19]</td><td>A366446</td></tr><tr><td>Commutative discrete aggregation functions</td><td>Prop. 3.2</td><td>[19]</td><td>A366447</td></tr><tr><td>Disc. agg. func. associative and quasitrivial</td><td>Prop. 3.3</td><td>[14, 17]</td><td>A293005</td></tr><tr><td>Disc. agg. func. associative, quasitrivial and neutral element</td><td>Prop. 3.3</td><td>[14, 17]</td><td>A002605</td></tr><tr><td>Disc. agg. func. associative, quasitrivial and absorbing element</td><td>Prop. 3.3</td><td>[14]</td><td>A293006</td></tr><tr><td>Disc. agg. func. associative, quasitrivial and with neutral and absorbing elements</td><td>Prop. 3.3</td><td>[14]</td><td>A293007</td></tr><tr><td>Disc. agg. func. bisymmetric and quasitrivial</td><td>Prop. 3.4</td><td>[28]</td><td>A296953</td></tr><tr><td>Disc. agg. func. bisymmetric, quasitrivial and neutral element</td><td>Prop. 3.4</td><td>[28]</td><td>A131577</td></tr><tr><td>Disc. agg. fun. bisymmetric, quasitrivial and absorbing element</td><td>Prop. 3.4</td><td>[28]</td><td>A296954</td></tr><tr><td>Smooth disc. agg. func. with absorbing element</td><td>Prop. 5.2</td><td>Sec. 5.1</td><td>A366449</td></tr><tr><td>Smooth disc. agg. func. with neutral element</td><td>Prop. 5.2</td><td>Sec. 5.1</td><td>A366449</td></tr><tr><td rowspan="10">Discrete conjunctions</td><td>Discrete conjunctions</td><td>Prop. 3.5</td><td>[19]</td><td>A360612</td></tr><tr><td>Commutative discrete conjunctions</td><td>Prop. 3.6</td><td>[19]</td><td>A366540</td></tr><tr><td>Discrete conjunctions with neutral element n</td><td>Prop. 3.7</td><td>[42]</td><td>A051255</td></tr><tr><td>Smooth discrete conjunctions</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Smooth and associative discrete conjunctions</td><td>Prop. 3.10</td><td>[20]</td><td>A000079</td></tr><tr><td>Commutative discrete conjunctions with neutral element n</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Idempotent and commutative discrete conjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A006125</td></tr><tr><td>Idempotent discrete conjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A053763</td></tr><tr><td>Discrete disjunctions</td><td>Prop. 3.5</td><td>[19]</td><td>A360612</td></tr><tr><td>Commutative discrete disjunctions</td><td>Prop. 3.6</td><td>[19]</td><td>A366540</td></tr><tr><td rowspan="8">Discrete disjunctions</td><td>Discrete disjunctions with neutral element 0</td><td>Prop. 3.7</td><td>[42]</td><td>A051255</td></tr><tr><td>Smooth discrete disjunctions</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Smooth and associative discrete disjunctions</td><td>Prop. 3.10</td><td>[20]</td><td>A000079</td></tr><tr><td>Commutative discrete disjunctions with neutral element 0</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Idempotent and commutative discrete disjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A006125</td></tr><tr><td>Idempotent discrete disjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A053763</td></tr><tr><td>Discrete quasi-copulas</td><td>Prop. 3.9</td><td>[20, 21]</td><td>A005130</td></tr><tr><td rowspan="3">Discrete copulas</td><td>Discrete copulas</td><td>Prop. 3.13</td><td>[18]</td><td>A000142</td></tr><tr><td>Commutative discrete copulas</td><td>Prop. 3.14</td><td>[18]</td><td>A000085</td></tr><tr><td>Associative discrete copulas</td><td>Prop. 3.15</td><td>[18]</td><td>A000079</td></tr><tr><td rowspan="2">Discrete t-norms</td><td>Smooth discrete t-norms</td><td>Prop. 3.16</td><td>[8]</td><td>A000079</td></tr><tr><td>Maximal Archimedean discrete t-norms</td><td>Prop. 3.17</td><td>[12]</td><td>A000045</td></tr><tr><td rowspan="2">Discrete t-conorms</td><td>Smooth discrete t-conorms</td><td>Prop. 3.16</td><td>[8]</td><td>A000079</td></tr><tr><td>Minimal Archimedean discrete t-conorms</td><td>Prop. 3.17</td><td>[12]</td><td>A000045</td></tr><tr><td rowspan="2">Discrete uninorms</td><td>Idempotent discrete uninorms with neutral element e</td><td>Thm. 3.5</td><td>[10]</td><td>—</td></tr><tr><td>Idempotent discrete uninorms</td><td>Thm. 3.5</td><td>[10, 14]</td><td>A000079</td></tr><tr><td rowspan="7">Discrete implications</td><td rowspan="7">Discrete uninorms having smooth underlying operators</td><td>Disc. uni. having smooth underlying operators, neutral element e, r + 1 idempotent elements on [0, e] and s + 1 idempotent elements on [e, n]</td><td>Thm. 3.7</td><td>[11]</td><td>—</td></tr><tr><td>Disc. uni. having smooth underlying operators and neutral element e</td><td>Thm. 3.7</td><td>[11]</td><td>—</td></tr><tr><td>Disc. uni. having smooth underlying operators</td><td>Thm. 3.7</td><td>[11]</td><td>A025192</td></tr><tr><td>Disc. uni. having smooth and idempotent or smooth and Archimedean underlying operators</td><td>Prop. 3.20</td><td>[11]</td><td>A366542</td></tr><tr><td>Disc. uni. having T<sub>L</sub> and S<sub>L</sub> as underlying operators.</td><td>Prop. 3.19</td><td>[11]</td><td>A005843</td></tr><tr><td>Disc. uni. having T<sub>M</sub> and S<sub>L</sub> as underlying operators.</td><td>Prop. 3.19</td><td>[11]</td><td>A000124</td></tr><tr><td>Disc. uni. having T<sub>M</sub> and S<sub>M</sub> as underlying operators.</td><td>Prop. 3.19</td><td>[11]</td><td>A000124</td></tr><tr><td rowspan="2">Discrete t-operators/ nullnorms</td><td>Smooth discrete t-operators/nullnorms</td><td>Prop. 3.21</td><td>[7]</td><td>A000027</td></tr><tr><td>Smooth and idempotent discrete t-operators/nullnorms</td><td>Prop. 3.21</td><td>[7]</td><td>A045623</td></tr><tr><td rowspan="7">Discrete implications</td><td>Discrete implications</td><td>Prop. 3.5</td><td>[19]</td><td>A360612</td></tr><tr><td>Smooth discrete implications</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Discrete implications satisfying (CP(N<sub>C</sub>))</td><td>Prop. 3.6</td><td>[19]</td><td>A366540</td></tr><tr><td>Discrete implications satisfying (NP)</td><td>Prop. 4.3</td><td>[19]</td><td>A367192</td></tr><tr><td>Discrete implications satisfying (CB)</td><td>Prop. 4.4</td><td>[19]</td><td>A367540</td></tr><tr><td>Discrete implications satisfying (OP)</td><td>Prop. 4.5</td><td>[19]</td><td>A367434</td></tr><tr><td>Discrete implications satisfying (IP)</td><td>Prop. 4.6</td><td>[19]</td><td>A367541</td></tr></tbody></table>

<caption>Table 12</caption>

Summary of the different enumerations considered in this paper, the result containing the formula, the original reference where it was proposed and the code associated to its entry in OEIS.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>Table 12</caption><thead><tr><th>Class</th><th>Subclass</th><th>Result</th><th>Original source</th><th>OEIS</th></tr></thead><tbody><tr><td rowspan="2">Discrete negations</td><td>Discrete negations</td><td>Prop. 2.1</td><td>[19]</td><td>A001700</td></tr><tr><td>Over anti-diagonal discrete negations</td><td>Prop. 2.2</td><td>[8]</td><td>A000108</td></tr><tr><td rowspan="14">Discrete aggregation functions</td><td>Discrete aggregation functions</td><td>Prop. 3.1</td><td>[19]</td><td>A366446</td></tr><tr><td>Commutative discrete aggregation functions</td><td>Prop. 3.2</td><td>[19]</td><td>A366447</td></tr><tr><td>Disc. agg. func. associative and quasitrivial</td><td>Prop. 3.3</td><td>[14, 17]</td><td>A293005</td></tr><tr><td>Disc. agg. func. associative, quasitrivial and neutral element</td><td>Prop. 3.3</td><td>[14, 17]</td><td>A002605</td></tr><tr><td>Disc. agg. func. associative, quasitrivial and absorbing element</td><td>Prop. 3.3</td><td>[14]</td><td>A293006</td></tr><tr><td>Disc. agg. func. associative, quasitrivial and with neutral and absorbing elements</td><td>Prop. 3.3</td><td>[14]</td><td>A293007</td></tr><tr><td>Disc. agg. func. bisymmetric and quasitrivial</td><td>Prop. 3.4</td><td>[28]</td><td>A296953</td></tr><tr><td>Disc. agg. func. bisymmetric, quasitrivial and neutral element</td><td>Prop. 3.4</td><td>[28]</td><td>A131577</td></tr><tr><td>Disc. agg. fun. bisymmetric, quasitrivial and absorbing element</td><td>Prop. 3.4</td><td>[28]</td><td>A296954</td></tr><tr><td>Smooth disc. agg. func. with absorbing element</td><td>Prop. 5.2</td><td>Sec. 5.1</td><td>A366449</td></tr><tr><td>Smooth disc. agg. func. with neutral element</td><td>Prop. 5.2</td><td>Sec. 5.1</td><td>A366449</td></tr><tr><td rowspan="10">Discrete conjunctions</td><td>Discrete conjunctions</td><td>Prop. 3.5</td><td>[19]</td><td>A360612</td></tr><tr><td>Commutative discrete conjunctions</td><td>Prop. 3.6</td><td>[19]</td><td>A366540</td></tr><tr><td>Discrete conjunctions with neutral element n</td><td>Prop. 3.7</td><td>[42]</td><td>A051255</td></tr><tr><td>Smooth discrete conjunctions</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Smooth and associative discrete conjunctions</td><td>Prop. 3.10</td><td>[20]</td><td>A000079</td></tr><tr><td>Commutative discrete conjunctions with neutral element n</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Idempotent and commutative discrete conjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A006125</td></tr><tr><td>Idempotent discrete conjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A053763</td></tr><tr><td>Discrete disjunctions</td><td>Prop. 3.5</td><td>[19]</td><td>A360612</td></tr><tr><td>Commutative discrete disjunctions</td><td>Prop. 3.6</td><td>[19]</td><td>A366540</td></tr><tr><td rowspan="8">Discrete disjunctions</td><td>Discrete disjunctions with neutral element 0</td><td>Prop. 3.7</td><td>[42]</td><td>A051255</td></tr><tr><td>Smooth discrete disjunctions</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Smooth and associative discrete disjunctions</td><td>Prop. 3.10</td><td>[20]</td><td>A000079</td></tr><tr><td>Commutative discrete disjunctions with neutral element 0</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Idempotent and commutative discrete disjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A006125</td></tr><tr><td>Idempotent discrete disjunctions</td><td>Prop. 5.4</td><td>Sec. 5.2</td><td>A053763</td></tr><tr><td>Discrete quasi-copulas</td><td>Prop. 3.9</td><td>[20, 21]</td><td>A005130</td></tr><tr><td rowspan="3">Discrete copulas</td><td>Discrete copulas</td><td>Prop. 3.13</td><td>[18]</td><td>A000142</td></tr><tr><td>Commutative discrete copulas</td><td>Prop. 3.14</td><td>[18]</td><td>A000085</td></tr><tr><td>Associative discrete copulas</td><td>Prop. 3.15</td><td>[18]</td><td>A000079</td></tr><tr><td rowspan="2">Discrete t-norms</td><td>Smooth discrete t-norms</td><td>Prop. 3.16</td><td>[8]</td><td>A000079</td></tr><tr><td>Maximal Archimedean discrete t-norms</td><td>Prop. 3.17</td><td>[12]</td><td>A000045</td></tr><tr><td rowspan="2">Discrete t-conorms</td><td>Smooth discrete t-conorms</td><td>Prop. 3.16</td><td>[8]</td><td>A000079</td></tr><tr><td>Minimal Archimedean discrete t-conorms</td><td>Prop. 3.17</td><td>[12]</td><td>A000045</td></tr><tr><td rowspan="2">Discrete uninorms</td><td>Idempotent discrete uninorms with neutral element e</td><td>Thm. 3.5</td><td>[10]</td><td>—</td></tr><tr><td>Idempotent discrete uninorms</td><td>Thm. 3.5</td><td>[10, 14]</td><td>A000079</td></tr><tr><td rowspan="7">Discrete implications</td><td rowspan="7">Discrete uninorms having smooth underlying operators</td><td>Disc. uni. having smooth underlying operators, neutral element e, r + 1 idempotent elements on [0, e] and s + 1 idempotent elements on [e, n]</td><td>Thm. 3.7</td><td>[11]</td><td>—</td></tr><tr><td>Disc. uni. having smooth underlying operators and neutral element e</td><td>Thm. 3.7</td><td>[11]</td><td>—</td></tr><tr><td>Disc. uni. having smooth underlying operators</td><td>Thm. 3.7</td><td>[11]</td><td>A025192</td></tr><tr><td>Disc. uni. having smooth and idempotent or smooth and Archimedean underlying operators</td><td>Prop. 3.20</td><td>[11]</td><td>A366542</td></tr><tr><td>Disc. uni. having T<sub>L</sub> and S<sub>L</sub> as underlying operators.</td><td>Prop. 3.19</td><td>[11]</td><td>A005843</td></tr><tr><td>Disc. uni. having T<sub>M</sub> and S<sub>L</sub> as underlying operators.</td><td>Prop. 3.19</td><td>[11]</td><td>A000124</td></tr><tr><td>Disc. uni. having T<sub>M</sub> and S<sub>M</sub> as underlying operators.</td><td>Prop. 3.19</td><td>[11]</td><td>A000124</td></tr><tr><td rowspan="2">Discrete t-operators/ nullnorms</td><td>Smooth discrete t-operators/nullnorms</td><td>Prop. 3.21</td><td>[7]</td><td>A000027</td></tr><tr><td>Smooth and idempotent discrete t-operators/nullnorms</td><td>Prop. 3.21</td><td>[7]</td><td>A045623</td></tr><tr><td rowspan="7">Discrete implications</td><td>Discrete implications</td><td>Prop. 3.5</td><td>[19]</td><td>A360612</td></tr><tr><td>Smooth discrete implications</td><td>Prop. 3.9</td><td>[20]</td><td>A005130</td></tr><tr><td>Discrete implications satisfying (CP(N<sub>C</sub>))</td><td>Prop. 3.6</td><td>[19]</td><td>A366540</td></tr><tr><td>Discrete implications satisfying (NP)</td><td>Prop. 4.3</td><td>[19]</td><td>A367192</td></tr><tr><td>Discrete implications satisfying (CB)</td><td>Prop. 4.4</td><td>[19]</td><td>A367540</td></tr><tr><td>Discrete implications satisfying (OP)</td><td>Prop. 4.5</td><td>[19]</td><td>A367434</td></tr><tr><td>Discrete implications satisfying (IP)</td><td>Prop. 4.6</td><td>[19]</td><td>A367541</td></tr></tbody></table>

Summary of the different enumerations considered in this paper, the result containing the formula, the original reference where it was proposed and the code associated to its entry in OEIS.