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# m-Stirling permutations, r-Whitney numbers and Lah numbers

Benjamin Randrianirina

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Benjamin Randrianirina. m-Stirling permutations, r-Whitney numbers and Lah numbers. 2025. hal-05091964

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Open licence - etalab
# m-Stirling permutations, r-Whitney numbers and Lah numbers

Benjamin RANDRIANIRINA*

Laboratoire de Mathématiques et Applications de l'Université de Fianarantsoa (LaMAF),
University of Fianarantsoa, Madagascar

## Abstract

We study the connections between m-Stirling permutations and r-Whitney numbers of the first kind. In particular, we generalize the notion of cycles in permutations to the case of m-Stirling permutations, calling them m-cycles, and show that the number of m-Stirling permutations having k m-cycles is $c_{m,0}(n, k)$, where $c_{m,r}(n, k)$ is the unsigned r-Whitney numbers. A new combinatorial interpretation of the unsigned r-Whitney numbers is then given. The case of the Stirling numbers of the second kind, particularly the combinatorial interpretation given by Mendez and Ramirez, is then revisited. We also study the convolution $C_{m,r}W_{m,s}$, where $C_{m,r} = (c_{m,r}(n, k))_{n,k \in \mathbb{N}}$, $W_{m,s} = (W_{m,s}(n, k))_{n,k \in \mathbb{N}}$. This allows us to define generalizations of Lah-numbers.

Keywords: r-Whitney numbers, m-Stirling permutations, composition of grammar, Lah number, convolution of matrices

## 1 Introduction

The r-Whitney numbers of the first and second kinds, introduced by Mezö in [14], can be defined respectively by the equations:

$$m^n x_{(n)} = \sum_{k=1}^{n} w_{m,r}(mx + r)^n \quad \text{and} \quad (mx + r)^n = \sum_{k=0}^{n} m^k W_{m,r}(n, k) x_{(k)} \quad (1)$$

where $x_{(n)} = x(x - 1) \cdots (x - n + 1)$. These numbers satisfy the recurrence relations:

$$w_{m,r}(n, k) = w_{m,r}(n - 1, k - 1) - (m(n - 1) + r)w_{m,r}(n - 1, k) \quad (2)$$

$$W_{m,r}(n, k) = W_{m,r}(n - 1, k - 1) + (km + r)W_{m,r}(n - 1, k) \quad (3)$$

with $w_{m,r}(0, 0) = w_{m,r}(0, 0) = 1$ and $W_{m,r}(n, k) = w_{m,r}(n, k) = 0$, if $k > n$

In [5], Cheong and Jung define an unsigned variant of the r-Whitney numbers of the first kind, which we denote by $(c_{m,r}(n, k))$, using the relation:

$$m^n x_{(n)} = \sum_{k=0}^{n} (-1)^{n-k} c_{m,r}(n, k) (mx + r)^k, \quad (4)$$

<footnote>*Email: benjamin.randrianirina@univ-fianarantsoa.mg</footnote>
satisfying the recurrence relation:

$$c_{m,r}(n, k) = c_{m,r}(n-1, k-1) + (r + (n-1)m)c_{m,r}(n-1, k). \quad (5)$$

Generalizing the Whitney numbers defined by Benoumhani in [1], several studies have been constructed on the r-Whitney numbers ([25], [22], [13], [6], [8], ...)

On the other hand, Lah numbers are numbers defined by Yvo Lah in [9] and satisfying:

$$L(n+1, k) = L(n, k-1) + (n+k)L(n, k). \quad (6)$$

In the literature, there exist some generalizations of Lah numbers: for instance, the r-Lah numbers introduced by Gabor Nyul and Gabriella Rác ([15]), which were later further generalized by Mark Shattuck in [24], the r-Whitney-Lah defined by Eszter Gyimesi and Gábor Nyul in [4], and the (l, r)-Lah Numbers defined by Aleks Zigon Tankosic in [26].

In [16], S. Park introduced the notion of m-multipermutations as a permutation $\sigma = \sigma_1\sigma_2\cdots\sigma_{mn}$ of the multiset $\{1^m, 2^m, \cdots, n^m\}$, such that for each triplet $(i, j, k)$ where $i < j < k$, if $\sigma_i = \sigma_k$ then $\sigma_i \le \sigma_j$. Generalizing the Stirling permutations defined by I. Gessel and R.P. Stanley in [7], this notion was later revisited by S. Jason, M. Kuba, and A. Panholzer in [21], calling it the m-Stirling permutation. We denote by $Q_m$ the L-species of m-Stirling permutations.

In this article, we will study, the connections between m-Stirling permutations and r-Whitney numbers of the first kind (section 3), and the r-Whitney numbers of the second kind (section 4). The convolution $C_{m,r}W_{m,r}$, where $C_{m,r} = (c_{m,r}(n,k))_{n,k \in \mathbb{N}}$ and $W_{m,r} = (W_{m,r}(n,k))_{n,k \in \mathbb{N}}$, allow us to define generalizations of Lah numbers in Section 4. In this study, we use the notion of grammar composition, studied in [20], in particular Theorem 5 concerning the combinatorial models of a composed grammar and Theorem 9 concerning the convolution of matrices associated with type (E) grammars. These results are recalled in Section 2.

For information on the theory of B-species and L-species, we refer the reader to [2], information about mixed species is provided in [19], and information about methods of resolution of system of differential equations see (see [10]).

**Notation:** If $\mathbb{F}$ and $\mathbb{G}$ are two L-species, the product of $\mathbb{F}$ and $\mathbb{G}$ will be denoted by $\mathbb{F} \cdot \mathbb{G}$, and their composition by $\mathbb{F} \circ \mathbb{G}$.

## 2 Grammar composition

Given a finite set $A = \{x_1, \cdots, x_n\}$ of variables, a Chen's grammar ([3]) over A is defined as a set of substitution rules replacing a letter in A by a formal function over A. In this article, a grammar is a map $G : A \to \mathbb{Z}[[A]]$ that allows replacing a variable $x_i$ with its image $G(x_i)$. We assume that $G(x_i) = f_i(x_1, \cdots, x_n)$ is the generating series of a asymmetric B-species $\mathbb{F}_i(X_1, \cdots, X_n)$, $1 \le i \le n$. We can associate to a grammar G:

1. an analytical differential operator: $\mathcal{G} = \sum_{i=1}^{n} f_i(x_1, \cdots, x_n) \frac{\partial}{\partial x_i}$

2. a combinatorial differential operator: $\mathcal{G} = \sum_{i=1}^{n} \mathbb{F}_i(X_1, \cdots, X_n) \frac{\partial}{\partial X_i}$

3. a combinatorial system of differential equations: $\{Y'_i = \mathbb{F}_i(Y_1, \cdots, Y_n), Y_i(0) = X_i, 1 \le i \le n\}$ whose combinatorial solution is the k-tuple $\overline{\mathcal{A}}_{\overline{\mathbb{F}}_i} = ((A_{\overline{\mathbb{F}}_i})_{i \in [k]})$, where $A_{\overline{\mathbb{F}}_i} = e^{TG}(X_i)$.
4. An analytical system of differential equations {$Y_i'(t) = F_i(Y_1(t), \dots, Y_n(t)), Y_i(0) = x_i, 1 \le i \le n$}

Solution of the system of differential equations is the n-tuple $\vec{A}_{\vec{F}} = ((A_{\vec{F},i})_{i \in [n]})$, of mixed species of $\vec{F}$-enriched rooted trees. And the generating series of the isomorphism types $A_{\vec{F},i} = T_{\vec{X}=\vec{x}}(A_{\vec{F},i})(1 \le i \le n)$, of these solutions are the solutions of the analytical system of differential equations (see [19] or [18]).

If $G$ is a grammar on $A = \{x_1, \dots, x_n\}$, and $F = F(X_1, \dots, X_n)$ a $B$-species, then the $L$-species of isomorphism types of $G(F)$ is denoted by $G(F)$. If $H$ and $K$ are two grammars over two disjoint alphabets $A$ and $B$, and $g \in \mathbb{R}[[B]]$, then we denote $Hg$ as the grammar over $A$ obtained by replacing each rule $u \to v$ in $H$ by $u \to vg$. The grammar $H_{K,g} = Hg \cup K$ over $A \cup B$ is called the composition of $H$ and $K$ in $g$ (see [20]).

Given $(H_i(Z_1, \dots, Z_k))_{i \in [k]}, (K_j(X_1, \dots, X_l))_{j \in [l]}, \text{ and } G = G(\vec{X})$, $B$-species whose generating series are $h_i = H_i(\vec{x})$, $k_j = K_j(\vec{x})$, and $g = G(\vec{x})$. We consider the grammar $H_{K,g} = Hg \cup K$, composed of the grammars $H = z_i \to h_i, i \in [k], K = x_j \to k_j, j \in [l]$ at $g$. We have the following theorem, shown in [20] Theorem 5:

**Theorem 1** For $1 \le i \le k$ and $1 \le j \le l$, we have:

$$e^{TH_{K,g}}(Z_i) = A_{\vec{H},i} \circ \left[ \int G((A_{\vec{R},j})_{j \in [l]}) \right] \qquad (7)$$

$$e^{TH_{K,g}}(X_j) = A_{\vec{K},j} \qquad (8)$$

More generally, for all $B$-species, $F = F_1(\vec{Z})F_2(\vec{X})$, we have:

$$e^{TH_{K,g}}(F) = \left[ (F_1 \circ \vec{A}_{\vec{H}}) \circ (\int G(\vec{A}_{\vec{K}})) \right] \cdot \left[ F_2 \circ \vec{A}_{\vec{K}} \right] \qquad (9)$$

$$\text{with } F_1 \circ \vec{A}_{\vec{H}} = F_1(A_{\vec{H},1}, \dots, A_{\vec{H},k}) = e^{TH}(F_1) \text{ and } F_2 \circ \vec{A}_{\vec{K}} = F_2(A_{\vec{K},1}, \dots, A_{\vec{K},l}) = e^{TK}(F_2)$$

We are interested by the grammar of type (E), which is grammar of the form: $H_g = \{z \to zg(\vec{x}), x_i \to h_i(\vec{x}), 1 \le i \le k\}$. This grammar is the composition of $H = \{z \to z\}$ with $K = \{x_i \to h_i(\vec{x}), 1 \le i \le k\}$ at $g(\vec{x})$. The isomorphism type of solutions satisfy $Hg = T_{\vec{X}=\vec{x}}e^{TH_g}(Z) = zE \circ [\int G(T_{\vec{X}=\vec{x}}\vec{A}_{\vec{H}})]$.

We consider two grammars of type (E): $H_g = \{z \to zg(\vec{x}), x_i \to h_i(\vec{x}), 1 \le i \le k\}$ and $K_{g_1} = \{v \to vg_1(\vec{u}), u_j \to k_j(\vec{u}), 1 \le j \le l\}$. The grammar $P_{\vec{g}} = \{v \to vg_1(\vec{u})g(\vec{x}), u_j \to k_j(\vec{u})g(\vec{x}), x_i \to h_i(\vec{x}), 1 \le i \le k, 1 \le j \le l\}$ is the composition of $H_1 = \{v \to v\}$ with $K_1 = \{u_j \to k_j(\vec{u})g(\vec{x}), x_i \to h_i(\vec{x}), 1 \le i \le k, 1 \le j \le l\}$ at $\vec{g}(\vec{u}, \vec{x}) = g_1(\vec{u})g(\vec{x})$. It is of type (E).

Let $E$ be the $L$-species of sets, $G$ an $L$-species such that $G(\emptyset) = \emptyset$, and $F = E \circ G$. If $s$ is an $F$-structure, each element of $s$ is called a connected component of $s$. The matrix $M = (a(n, k))_{n,k \in \mathbb{N}}$, where $a(n, k)$ is the number of elements in $F[n]$ having $k$ connected components, is called the matrix of $F$. It is shown in [20], theorem 8 that:

**Theorem 2** Let $M = (a(n, k))_{n, k \in \mathbb{N}}$, $N = (b(n, k))_{n, k \in \mathbb{N}}$ and $\lambda = (\lambda(n, k))_{n, k \text{ be the matrices of } Hg, Kg_1, \text{ and } P\vec{g}}$. Then, $\lambda = MN$. In other words:

$$\lambda(n, k) = \sum_{i \ge 0} a(n, i)b(i, k) \qquad (10)$$
More generally, if $\mathbb{F} = \mathbb{F}_1(\overrightarrow{X})\mathbb{F}_2(\overrightarrow{U})$ is a $\mathbb{B}$-species, $\lambda_{\mathbb{F}} = (\lambda_{\mathbb{F}}(n, k))$, $\lambda_{\mathbb{F}_1} = (\lambda_{\mathbb{F}_1}(n, k))$ and $\lambda_{\mathbb{F}_2} = (\lambda_{\mathbb{F}_2}(n, k))$ are the matrices of $\mathbb{P}_{\overline{\mathbb{g}}}(v\mathbb{F})$, $\mathbb{H}_g(z\mathbb{F}_1)$, and $\mathbb{K}_{g_1}(v\mathbb{F}_2)$, then:

$$ \lambda_{\mathbb{F}} = \lambda_{\mathbb{F}_1} \lambda_{\mathbb{F}_2}. $$

### 3 $m$-Stirling permutations and $r$-Whitney numbers of the first kind.

Consider the one-variable grammar $G := \{x \to x^{m+1}\}$, whose associated combinatorial differential equation is:

$$ Y' = Y^{m+1}, \quad Y(0) = X \qquad (11) $$

The combinatorial solution of this equation is the mixed species $e^{TG}(X) = A_{m+1}(T, X)$ of complete $(m+1)$-ary trees, where leaves are of type $X$ (see [20]). The isomorphism types with respect to $X$ is the $\mathbb{L}$-species $\mathbb{A}_{m+1,\alpha}(T) = T_{X=x}e^{TG}(X)$ of increasing $(m+1)$-ary trees, weighted by $\alpha(b) = x^{mn+1}$ if $b$ has $n$ internal vertices. In [16], S. Park show that there exists a bijection between increasing $(m+1)$-ary trees and $m$-Stirling permutations (see also [21], [12]). This bijection can be represented by Figure 1. The $\mathbb{L}$-species $\mathbb{A}_{m+1,\alpha}$

<figure><img src="image_3.png" /><figcaption>Figure 1: Bijection between $\mathbb{A}_{m+1}$ and $\mathcal{Q}_m$</figcaption></figure>

is therefore isomorphic to the $\mathbb{L}$-species $\mathcal{Q}_{m,\alpha}$ of $m$-Stirling permutations weighted by $\alpha(\sigma) = x^{mn+1}$ if $\alpha \in \mathcal{Q}_m[n]$.

The exponential generating series, obtained by solving the differential equation $y'(t) = y^{m+1}(t)$, $y(0) = x$, is:

$$ \mathbb{A}_{m+1,\alpha}(t) = \sum_{n \ge 0} \mathcal{G}^n(x) \frac{t^n}{n!} = x(1 - mx^m t)^{-\frac{1}{m}}, \quad \text{if } m \ge 1, \text{ and } \mathbb{A}_{1,\alpha}(t) = xe^t. \qquad (12) $$

It's clear that $\mathcal{G}^n(x) = (m+1)(2m+1)\cdots(m(n-1)+1)x^{mn+1}$. We can deduce Park's result ([16]), which states that the number of $m$-Stirling permutations on $[n]$ is:

$$ |\mathcal{Q}_m[n]| = (m+1)(2m+1)\cdots(m(n-1)+1). \qquad (13) $$

Now, consider the following grammar, where $\mu$ is a formal variable, counter of connected components:

$$ H_g = \{z \to \mu z x^m, x \to x^{m+1}\} \qquad (14) $$

who is the composition of the grammar $H = \{z \to \mu z\}$ and $K = \{x \to x^{m+1}\}$ in $g(x) = x^m$. We have $e^{TH}(Z) = Z\mathbb{E}_\beta(T)$, where $\mathbb{E}_\beta$ is the $\mathbb{L}$-species of sets weighted by $\beta(A) = \mu^{|A|}$, and that $e^{TK}(X) = A_{m+1}(T, X)$. So $e^{TH_g}(Z) = Z\mathbb{E}_\beta \circ \int A_{m+1}(T, X)$. We deduce that:

$$ T_{Z=z, X=x}(e^{TH_g}(Z)) = \mathbb{H}_g = z\mathbb{E}_\beta \circ \int \mathbb{A}_{m+1,\alpha}^m = z\mathbb{E}_\beta \circ \int \mathcal{Q}_{m,\alpha}^m \qquad (15) $$
Equation (15) shows that a connected component of a $\mathbb{H}_g$-structure is an $\int Q_{m\alpha}^m$-structure. That is an $m$-cycle in the following sense:

**Definition 1** An element $\sigma = \sigma_1\sigma_2\cdots\sigma_{mn}$ of $Q_m[n]$ is called an $m$-cycle if $\sigma_1 = 1$. An $m$-cycle of $\sigma = \sigma_1\sigma_2\cdots\sigma_{mn}$ is a subsequence $\sigma' = \sigma_i\sigma_{i+1}\cdots\sigma_s$ of $\sigma$ such that: $\sigma'$ is an $m$-Stirling permutation on a subset $A$ of $[n]$, $\sigma_i = \min A$, and $\sigma_i > \sigma_{s+1}$.

Thus, for $\sigma = 6624423311$, $244233$ is an $m$-cycle of $\sigma$, but $2442$ and $24423311$ are not.

**Lemma 1** Every $m$-Stirling permutation can be uniquely decomposed into a set of disjoint $m$-cycles.

<figure><img src="image_4.png" /><figcaption>Figure 2: Decomposition of an $m$-Stirling permutation</figcaption></figure>

Figure 2 represents the decomposition of the 2-Stirling permutation from Figure 1. To obtain this, one can use the notion of left-to-right minima (([16] or [12]): An element $i$ of $[mn]$ is a left-to-right minimum of $\sigma = \sigma_1\sigma_2\cdots\sigma_{mn}$, if $i = 1$ or $\forall j, j < i \to \sigma_j > \sigma_i$. Thus, left-to-right minima of $\sigma = 5512214433$ are 5 and 1, and $\sigma$ decomposes as $\sigma = \{[55], [12214433]\}$. In the same way, the decomposition of $\sigma = 6624423311$ is $\{[66], [244233], [11]\}$.

Therefore, we have:

$$ \mathbb{H}_g = z Q_{m,\beta} \qquad (16) $$

where $Q_{m,\beta}$ is the $\mathbb{L}$-species of $m$-Stirling permutations weighted by $\beta(\sigma) = \mu^k x^{mn}$ if $\sigma$ has $k$ $m$-cycles. Knowing that $\mathbb{E}_{\beta}(t) = e^{\mu t}$, equation (12) give us:

$$ \mathbb{H}_g(t) = z(1 - mx^m t)^{-\frac{\mu}{m}} \quad \text{and} \quad Q_{m,\beta}(t) = (1 - mx^m t)^{-\frac{\mu}{m}} \qquad (17) $$

Thus, we obtain a result by Park on the enumeration of the $r$-Stirling permutations according to the left-right minima (([16], proposition 1.13).

**Theorem 3** Let $c_m(n, k)$ the numbers of $m$-Stirling permutations having $k$ $m$-cycles. We have $\mathcal{H}_g^n(z) = \sum_{k=1}^n c_m(n, k)\mu^k zx^{mn}$, and the sequence $(c_m(n, k))$ satisfies the recurrence relation:

$$ c_m(n, k) = c_m(n - 1, k - 1) + m(n - 1)c_m(n - 1, k) \qquad (18) $$

with initial conditions $c_m(0, 0) = 1$, and $c_m(n, 0) = c_m(0, k) = 0$ if $n \ge 1$. In other words,

$$ c_m(n, k) = c_{m,0}(n, k), \qquad (19) $$

where $c_{m,r}(n, k)$ are the unsigned $r$-Whitney numbers of first kind.

*Proof:* Indeed, an $(TH_g)^n(Z)$-structure is obtained by applying $(TH_g)$ to an $(TH_g)^{n-1}(Z)$-structure. It consists of:

* Either replacing the leaf $z$ with a node of type $T$, weighted by $\mu$ and labeled $n$, followed by a leaf labeled $z$ and $m$ leaves $x$. We obtain a new $m$-cycle with weight $\mu z x^m$.
* Or replacing a leaf $x$ with a node of type $T$, labeled $n$, followed by $m+1$ leaves $x$. This results in a structure with the same number of $m$-cycles but with $m$ additional leaves, each weighted by $x$.

Since a tree on $[n-1]$ contains exactly $m(n-1)$ leaves weighted by $x$, we obtain the desired result. ◇

**Corollary 4** We have:

$$ \sum_{k=0}^{n} c_m(n, k)\mu^k = \mu(\mu + m)\cdots(\mu + m(n-1)), \qquad (20) $$

*Proof:* Note that Equation (18) is equivalent to $\mathcal{H}_g^n(z) = \mathcal{H}_g(\mathcal{H}_g^{n-1}(z)) = (\mu + m(n-1))x^m\mathcal{H}_g^{n-1}(z)$. ◇

Noticing that $\mu(\mu + m)\cdots(\mu + m(n-1)) = m^n \frac{\mu}{m} (\frac{\mu}{m} + 1) \cdots (\frac{\mu}{m} + n - 1)$, we obtain theorem 6 of [12]:

**Corollary 5**

$$ c_m(n, k) = m^{n-k}c(n, k) \qquad (21) $$

where $c(n, k)$ is the unsigned Stirling number of first kind, that is, $c(n, k) = c_1(n, k)$ is the number of permutations of $[n]$ having $k$ cycles.

Let $\mathbb{F}(X) = X^r$. We have $e^{\mathbb{F}}(\mathbb{F}) = e^{\mathbb{F}}(\mathbb{F}) = (e^{\mathbb{F}}(X))^r = (A_{m+1}(T, X))^r$. Thus $e^{\mathbb{F}}(\mathbb{F}) = [Z\mathbb{E}_\beta(\int(\mathbb{A}_{m+1}(T, X))^m)] \cdot (\mathbb{A}_{m+1}(T, X))^r$. We can deduce that:

$$ \mathbb{H}_g(z\mathbb{F}) = z\mathbb{Q}_{m,\beta} \cdot (\mathbb{Q}_{m,\alpha})^r. \qquad (22) $$

Equations (12) and (17) show that:

$$ \mathbb{H}_g(z\mathbb{F})(t) = \sum_{n \ge 0} \mathcal{H}_g^n(zx^r) \frac{t^n}{n!} = \frac{zx^r}{(1 - mx^m t)^{\frac{r+\mu}{m}}} \qquad (23) $$

A $\mathbb{H}_g(z\mathbb{F})$-structure on $[n]$ is therefore a pair $s = (s_1, s_2)$, where $s_1$ is a $\mathbb{Q}_{m,\beta}$-structure on a subset $A$ of $[n]$, weighted by $\beta(s_1) = \mu^k kzx^{m|A|}$ if $s_1$ has $k$ $m$-cycles, and $s_2$ is a $\mathbb{Q}_m$-structure on $[n] \setminus A$, with weight $x^{m(n-|A|)+r}$. Setting $\mathcal{H}_g^n(zx^r) = \sum_{k=1}^n \lambda_{m,r}(n, k)\mu^k zx^{mn+r}$, and writing $\mathcal{H}_g^{n+1}(zx^r) = \mathcal{H}_g(\mathcal{H}_g^n(zx^r))$, we have $\lambda_{m,r}(n+1, k) = \lambda_{m,r}(n, k-1) + (mn+r)\lambda_{m,r}(n, k)$. Applying (5), we obtain $c_{m,r}(n, k) = \lambda_{m,r}(n, k)$. We deduce the following theorem:

**Theorem 6** The unsigned $r$-Whitney numbers of the first kind $c_{m,r}(n, k)$ satisfy:

$$ \mathcal{H}_g^n(zx^r) = \sum_{k=0}^{n} c_{m,r}(n, k)\mu^k zx^{mn+r}, \qquad (24) $$

that is, $C_{m,r} = (c_{m,r}(n, k))$ is the matrix of $\mathbb{H}_g(z\mathbb{F})$.

**Corollary 7** The polynomial $C_{m,r}(n, \mu) = \sum_{k=0}^n c_{m,r}(n, k)\mu^k$ satisfy:

$$ C_{m,r}(n, \mu) = (\mu + r)(\mu + r + m)\cdots(\mu + r + m(n - 1)) \qquad (25) $$

*Proof:* Indeed, the relation $\mathcal{H}_g^{n+1}(zx^r) = \mathcal{H}_g(\mathcal{H}_g^n(zx^r))$ can be translated as $\mathcal{H}_g^{n+1}(zx^r) = (\mu + r + mn)x^m\mathcal{H}_g^n(zx^r)$. ◇
**Corollary 8** The r-Whitney numbers ($c_{m,r}(n, k)$) satisfy:

$$c_{m,r}(n, k) = \sum_{i=k}^{n} \binom{i}{k} m^{n-i} r^{i-k} c(n, i) = \sum_{i=k}^{n} \binom{i}{k} r^{i-k} c_m(n, i) \quad (26)$$

*Proof:* Indeed $(\mu+r)(\mu+r+m) \cdots (\mu+r+m(n-1)) = m^n \frac{\mu+r}{m} (\frac{\mu+r}{m}+1) \cdots (\frac{\mu+r}{m}+(n-1)) = \sum_{i=0}^{n} m^{n-i} c(n, i) (\mu+r)^i = \sum_{i=0}^{n} m^{n-i} c(n, i) \sum_{k=0}^{i} \binom{i}{k} \mu^k r^{i-k}$. $\diamond$

Let's reconsider the grammar $K = \{x \to x^{m+1}\}$. One can verify that $K^n(x^r) = r(m+r) \cdots (m(n-1)+r)x^{mn+r}$. Thus, $K^n(x^r) = \sum_{k=0}^{n} m^{n-k} c(n, k) r^k x^{mn+s}$. Therefore, if $\mathbb{F}(X) = X^r$, we have:

**Theorem 9** The number of $K(\mathbb{F})$-structures on $[n]$ is:

$$|K(\mathbb{F})[n]| = |(Q_m)^r[n]| = \sum_{k=0}^{n} m^{n-k} c(n, k) r^k. \quad (27)$$

Relation (22) shows that: $c_{m,r}(n, k) = \sum_{j=k}^{n} \binom{n}{j} c_m(j, k) |(Q_m)^r[n-j]|$. Hence:

$$c_{m,r}(n, k) = \sum_{j=k}^{n} \sum_{i=0}^{n-j} \binom{n}{j} c(j, k) c(n-j, i) m^{n-k-i} r^i \quad (28)$$

By applying Leibniz's formula to $zx^r = zx^s x^{r-s}$, we obtain:

$$c_{m,r}(n, k) = \sum_{k=j}^{n} \binom{n}{j} c_{ms}(k, j) \left( \sum_{i=0}^{k} c_m(n-k, i) (r-s)^i \right) \quad (29)$$

Case of r-Whitney numbers of the first kind $w_{m,r}(n, k)$ is easily deduced from the relation $w_{m,r}(n, k) = (-1)^{n-k} c_{m,r}(n, k)$. Using the following grammar,

$$H_{1g} = \{z \to \mu z x^m, x \to -x^{m+1}\}. \quad (30)$$

and taking $\mathbb{F}(X) = X^r$, we obtain a combinatorial model:

$$\mathcal{H}_{1g}(z\mathbb{F}) = z Q_{m,\beta'} \cdot (Q_{m,\alpha'})^r \quad (31)$$

where $Q_{m,\beta'}$ (respectively $Q_{m,\alpha'}$) is the $\mathbb{L}$-species of $m$-Stirling permutations weighted by $\beta'(\sigma) = (-1)^{n-k} \mu^k x^{mn}$ (respectively $\alpha'(\sigma) = (-1)^{n-1} \mu^k x^{mn+1}$) if $\sigma \in Q_m[n]$ and $\sigma$ has $k$ $m$-cycles. The generating series is:

$$\mathbb{H}_{1,g}(z\mathbb{F})(t) = zx^r(1 + mx^m t)^{\frac{\mu-r}{m}} \quad (32)$$

By setting $\mathcal{H}_{1,g}(z) = \sum_{k=0}^{n} w_m(n, k) \mu^k z x^m$, we have:

$$w_m(n, k) = w_{m,0}(n, k) = (-1)^{n-k} c_m(n, k) = m^{n-k} s(n, k) \quad (33)$$

where $s(n, k)$ are the Stirling numbers of the first kind. Furthermore,

$$\sum_{k=0}^{n} w_m(n, k) \mu^k = \mu(\mu - m) \cdots (\mu - m(n - 1)) \quad (34)$$

More generally, the first-kind Whitney r-numbers ($w_{m,r}(n, k)$) m, satisfy:

$$\sum_{k=0}^{n} w_{m,r}(n, k) \mu^k = (\mu - r)(\mu - r - m) \cdots (\mu - r - m(n - 1)) \quad (35)$$

$$w_{m,r}(n, k) = \sum_{k=0}^{n} \sum_{i=k}^{n} \binom{i}{k} m^{n-i} r^{i-k} (-1)^{i-k} s(n, i) \quad (36)$$
## 4 The r-Whitney number of the second kind

Let us consider the grammar $H_{2,g} := \{z \to zx^m, x \to x\}$. In ([17]), Hao and all show that $\forall n \in \mathbb{N}$, $\mathcal{H}_{2,g}^n(zx) = z \sum_{k=0}^n W_{m,1}(n, k)x^{mk+1}$, where $W_{m,1}(n, k)$ is Whitney number of the second kind (Benoumhani[1]). In ([11]), Mendez and Ramirez show that $\mathcal{H}_{2,g}^n(zx^r) = \sum_{k=0}^n W_{m,r}(n, k)zx^{mk+r}$. We modify this grammar as follow:

$$H_{2,g} := \{z \to \mu zx^m, x \to x\}. \qquad (37)$$

In this way, we will have the r-Dowling polynomials, defined by Mark Shattuck in [23].

**Theorem 10** The isomorphism type of solution of the system of differential equations associated with the grammar $H_{2,g}$ is:

$$\mathbb{H}_{2,g} = z\mathbb{E}_{\beta} \circ \left( \int (x\mathbb{E})^m \right), \qquad (38)$$

More generally, if we set $\mathbb{F}(X) = X^r$, then we have:

$$\mathbb{H}_{2,g}(z\mathbb{F}) = z \left( \mathbb{E}_{\beta} \circ \int (x\mathbb{E})^m \right) \cdot (x\mathbb{E})^r \qquad (39)$$

The exponential generating series is:

$$\mathcal{H}_{2,g}(t) = ze^{\mu \frac{x^m}{m}(e^{mt}-1)} \quad \text{and} \quad \mathcal{H}_{2,g}(z\mathbb{F})(t) = zx^r e^{rt}e^{\mu \frac{x^m}{m}(e^{mt}-1)} \qquad (40)$$

*Proof:* The grammar $H_{2,g}$ is the composition of $H = \{z \to \mu z\}$ with $K = \{x \to x\}$ in $g(x) = x^m$. It is clear that $e^{TH}(Z) = Z\mathbb{E}_{\beta}(T)$ et $e^{TK}(X) = X\mathbb{E}(T)$. Theorem 1 give us the result. ◇

Mendez and Ramirez's result means that the matrix of $\mathbb{H}_{2,g}(z\mathbb{F})$ is $W_{m,r} = (W_{m,r}(n, k))_{n,k}$. For $r = 0$, we have:

**Theorem 11** The number $W_m(n, k) = W_{m,0}(n, k)$ of $\mathbb{H}_{2g}$-structures on $[n]$ with $k$ connected component satisfy:

$$W_m(n, k) = m^{n-k}S(n, k) \qquad (41)$$

where $S(n, k)$ is the Stirling number of second kind.

*Proof:* Equation (38) shows that an $\mathbb{H}_{2g}$-structure on $[n]$ with $k$ connected components is obtained by considering a partition $\pi = \{b_1, b_2, \dots, b_k\}$ of $[n]$ into $k$ blocks, and constructing on each $b_i$ an $\int (x\mathbb{E})^m$-structure, that is, an $(x\mathbb{E})^m$-structure on $b_i \setminus \min b_i$. Such a structure corresponds to a mapping from $b_i \setminus \min b_i$ to $[m]$. ◇

Using equation (39), we deduce Theorem 3.2 of [5] from (41):

$$W_{m,r}(n, k) = \sum_{j=0}^{n-k} \binom{n}{j} r^{n-j} W_m(j, k) = \sum_{j=0}^{n-k} \binom{n}{j} r^{n-j} m^{j-k} S(j, k) \qquad (42)$$

The r-Dowling polynomials ([23]) are the polynomials $\mathcal{D}_{m,r}(n, \mu) = \sum_{k=0}^n W_{m,r}(n, k)\mu^k = \mathcal{H}_{2,g}^n(zx^r)|_{z=x=1}$. We have $\mathcal{H}^{n+1}(zx^r) = \mathcal{H}_{2,g}(\mathcal{H}_{2,g}^n(zx^r)) = \mathcal{H}_{2,g}(\sum_{k=0}^n W_{m,r}\mu^k zx^{mk+r}) = \sum_{k=0}^n W_{m,r}\mu^{k+1}zx^{mk+r}) + \sum_{k=0}^n (mk+r)W_{m,r}\mu^kzx^{mk+r})$, then $\mathcal{D}_{m,r}(n+1, \mu) = \sum_{k=0}^n W_{m,r}\mu^{k+1} + \sum_{k=0}^n (mk+r)W_{m,r}\mu^k$. We can deduce that:

$$\mathcal{D}_{m,r}(n+1, \mu) = (\mu + r)\mathcal{D}_{m,r}(n, \mu) + m\mu \frac{d\mathcal{D}_{m,r}(n, \mu)}{d\mu} \qquad (43)$$

On the other hand, Leibniz's formula shows that: $\mathcal{H}_{2,g}(zx^r) = \sum_{j=0}^n \binom{n}{j} \mathcal{H}_{2,g}^j(z) \mathcal{H}_{2,g}^{n-j}(x^r)$. We have $\mathcal{H}_{2,g}^{n-j}(x^r) = r^{n-j}x^r$ and $\mathcal{H}_{2,g}^j(z) = \sum_{k=0}^j W_m(j, k)\mu^k x^{mj}$.
**Theorem 12** The r-Dowling polynomials satisfy:

$$ \mathcal{D}_{m,r}(n, \mu) = \sum_{k=0}^{n} \sum_{j=k}^{n} \binom{n}{j} r^{n-j} W_m(j, k) \mu^k, \quad (44) $$

More generally, we have $(\mathcal{H}_{2,g})^n(x^r y) = \sum_{k=0}^n \binom{n}{k} (\mathcal{H}_{2,g})^k (x^s y) (\mathcal{H}_{2,g})^{n-k} (x^{r-s})$, and $(\mathcal{H}_{2,g})^{n-k} (x^{r-s}) = (r-s)^{n-k} x^{r-s}$. Thus

$$ \mathcal{D}_{m,r}(n, \mu) = \sum_{k=0}^{n} \binom{n}{k} (r-s)^{n-k} \mathcal{D}_{m,r}(k, \mu), \quad (45) $$

that is equivalent to:

$$ W_{m,r}(n, k)) = \sum_{l=k}^{n} \binom{n}{l} (r-s)^{n-l} W_{m,s}(l, k). \quad (46) $$

# 5 Lah-numbers and Generalizations

Using the concept of grammar composition, we will define two generalizations of Lah-numbers. In this way, we generalize results obtained in [20] Example 7 and 8 in which we show that the matrix associated with the grammar $H_{K,g}^* = \{z \to \mu zx^2, x \to x^2\}$ is the matrix $L = (L(n, k)_{n,k})$ of Lah-numbers $L(n, k)$. Let us recall that the $\mathbb{L}$-species $\mathbb{H}_{K,g}^*(z\mathbb{F})$, where $\mathbb{F}(X) = X^r$, gives us a generalization of the Lah numbers: that is the matrix $(L_r(n, k))_{n,k}$ associated with $\mathbb{H}_{K,g}^*(z\mathbb{F})$, satisfying the recurrence relation:

$$ L_r(n+1, k) = L_r(n, k-1) + (n+k+r)L_r(n, k). \quad (47) $$

Note that

$$ (x+r)^{(n)} = \sum_{k=\oplus}^{n} L_r(n,k) x_{(k)}, \quad (48) $$

where $x^{(n)} = x(x+1)\cdots(x+n-1)$ and $x_{(k)} = x(x-1)\cdots(x-k+1)$. The numbers $L_{2r}(n, k)$, is the r-Lah numbers, studied by G. Nyul and G. Rácz in [15] and by M Shattuck in [24].

Let us also recall that the matrices associated with the grammars $H_g = \{z \to \mu zx, x \to x^2\}$ and $K_{g_1} = \{v \to vu, u \to u\}$ are respectively $C = (c(n, k))_{n,k}$, where $c(n, k)$ is the unsigned Stirling numbers of the first kind and $S = (S(n, k))_{n,k}$ where $S(n, k)$ is the Stirling numbers of the second kind. Using the grammar $P_{\bar{g}} = \{v \to vux, u \to ux, x \to x^2\}$, which is the composition of $H_g$ and $K_{g_1}$ in $g(x) = x$, we show that $L = CS$, that is $L(n, k) = \sum_{i=k}^n c(n, i)S(i, k)$. This grammar allow us to define an other generalization of the Lah-numbers, who is the matrix $(L_{r,s}(n, k))_{n,k}$ of $P_{\bar{g}}(v\mathbb{F})$, where $\mathbb{F}(U, X) = U^r X^s$, satisfying the recurrence relation:

$$ L_{r,s}(n+1, k) = L_{r,s}(n, k-1) + (n+k+r+s)L_{r,s}(n, k). \quad (49) $$

It's clear that for all $n, k$, $L_{r,s}(n, k) = L_{r+s}(n, k)$ and $L_{r,s}(n, k) = L_{s,r}(n, k)$. Notice that:

$$ \mathbb{P}_{\bar{g}}(v\mathbb{F}) = (v\mathbb{E} \circ \mathbb{L}_{\gamma}^{+}) \cdot (\mathbb{L}_{\gamma})^{r} \cdot (\mathbb{L}_{\alpha})^{s}, \quad (50) $$

where $\mathbb{L}_{\gamma}$ is the $\mathbb{L}$-specie weighted by $\gamma(\sigma) = ux^n$ if $\sigma \in \mathbb{S}[n]$, and $\mathbb{L}_{\alpha}$ is the $\mathbb{L}$-species weighted by $\alpha(\sigma) = x^{n+1}$.
In the following, we will generalize these results and propose some others extensions of Lah numbers.

The matrices associated with the grammars $H_g = \{z \to \mu z x^m, x \to x^{m+1}\}$ and $K_{g_1} := \{v \to \mu v u^m, u \to u\}$ are respectively $C_m = (c_m(n, k))$ and $W_m = (W_m(n, k))$. Then the matrix $\Lambda_m = (\Lambda_m(n, k))_{n,k}$ associated with the grammar

$$ M_{\bar{g}} = \{v \to \mu v u^m x^m, u \to u x^m, x \to x^{m+1}\} \quad (51) $$

verify $\Lambda_m = C_m W_m$, that is:

$$ \Lambda_m(n, k) = \sum_{i=k}^{n} c_m(n, i) W_m(i, k). \quad (52) $$

On the other hand, Theorem 1 shows that the solutions of the combinatorial differential system associated with this grammar are:

$$ M_{\bar{g}} = v \mathbb{E}_{\beta} \circ \left( \left( \int (u \mathbb{E})^m \right) \circ \int (Q_{m,\alpha})^m \right). \quad (53) $$

Knowing that $Q_{m,\alpha}(t) = \frac{x}{(1-mx^m t)^{\frac{1}{m}}}$, we have $(Q_{m,\alpha})^m(t) = \frac{x^m}{(1-mx^m t)^m}$. Therefore $(\int (Q_{m,\alpha})^m)(t) = \int_0^t \frac{x^m}{(1-mx^m s)^{\frac{1}{m}}} ds = \ln(\frac{1}{(1-mx^m t)^{\frac{1}{m}}})$. In the other hand, we have $(\int (u \mathbb{E})^m)(t) = \int_0^t u^m e^{ms} ds = \frac{u^m}{m}(e^{mt} - 1)$ et $\mathbb{E}_{\beta}(t) = e^{\mu t}$, therefore $(v \mathbb{E}_{\beta} \circ \int (u \mathbb{E})^m)(t) = v e^{\frac{\mu u^m}{m}(e^{mt}-1)}$. Thus

$$ M_{\bar{g}}(t) = v \exp\left(\frac{\mu u^m x^m t}{1 - m x^m t}\right) \quad (54) $$

The relation (53) also shows that each $M_{\bar{g}}$-structure on $[n]$ corresponds to a pair $s = (s_1, s_2)$, where $s_1$ is a $\mathbb{E} \circ (\int (\mathbb{E})^m)$-structure on $[i]$, weighted by $\mu^k v^{mk}$ if it has $k$ connected components, and $s_2$ is an $\mathbb{E} \circ \int (\mathcal{Q}m)^m$-structure on $[n]$ with $i$ $m$-cycles, weighted by $x^{mn}$. Thus, an $M_{\bar{g}}$-structure on $[n]$ with $k$ connected components has total weight $v \mu^k u^{mk} x^{mn}$. Thus, $\Lambda_m(n, k)$ is the number of $M_{\bar{g}}$-structures on $[n]$ such that $s_1$ has $k$ connected components, that is, $M_{\bar{g}}^n(v) = \sum_{k=1}^n \Lambda_m(n, k) \mu^k v u^{mk} x^{mn}$. The relation $M_{\bar{g}}^{n+1}(v) = M_{\bar{g}}(M_{\bar{g}}^n(v))$ is then expressed by the recurrence relation:

$$ \Lambda_m(n+1, k) = \Lambda_m(n, k-1) + m(n+k)\Lambda_m(n, k) \quad (55) $$

**Proposition 1** The numbers $\Lambda_m(n, k)$ satisfy:

$$ \Lambda_m(n, k) = m^{n-k} L(n, k) \quad (56) $$

where $L(n, k)$ are the Lah numbers.

*Proof:* We prove this proposition in two different ways. The grammar used in the second proof will allow us to define an other generalization of Lah numbers.

*First proof* Using (52), (21) and (41), we have $\Lambda_m(n, k) = \sum_{i=k}^n m^{n-i} c(n, i) m^{i-k} W(i, k)$.

*Second proof:* Let us consider the following grammar:

$$ H_m = z \to \mu z x^2, x \to mx^2. \quad (57) $$

It is clear that $H_m = z \mathbb{E}_{\beta} \circ (A_{2,\gamma})^+ = z \mathbb{E}_{\beta} \circ \mathbb{L}_{\gamma}^+$, where $A_{2,\gamma}$ is the $\mathbb{L}$-species of increasing binary trees weighted by $\gamma(b) = m^{n-1}x^{n+1}$, if $b \in A_2[n]$, and $\mathbb{L}_{\gamma}$ is the species of permutations
<figure><img src="image_5.png" /><figcaption>Figure 3: A set of increasing binary trees with weight $m^6 \mu^2 z x^{10}$</figcaption></figure>

weighted by $\gamma(\sigma) = m^{n-1}x^{n+1}$ if $\sigma \in \mathbb{L}[n]$. The weight of an $\mathbb{H}_g$-structure with $k$ connected components is $m^{n-k}\mu^kzx^{n+k}$. Let us define $\mathcal{H}_g^n(z) = \sum_{k=1}^n a_m(n, k)m^{n-k}\mu^kzx^{n+k}$. Then the sequence $(a_m(n, k))$ satisfies the recurrence relation $a_m(n+1, k) = a_m(n, k-1)+m(n+k)a_m(n, k)$. Therefore, $a_m(n, k) = \Lambda_m(n, k)$. Figure 3 is an example of an $\mathbb{H}_g$-structure (with $k=2$) isomorphic to the set of lists {2846, 5137} with weight $m^6\mu^2zx^{10}$. Noting that an $\mathbb{H}_m$-structure on $[n]$ with $k$ connected components is a set of $k$ increasing binary trees, i.e. a set of $k$ lists, in which every point is weighted by $m$ except for the root of each connected component who is weighted by $\mu$. It's clear that $\Lambda_m(n, k) = m^{n-k}L(n, k)$. ◇

The grammar $M_{\bar{g}} = \{v \to \mu vu^m x^m, u \to ux^m, x \to x^{m+1}\}$ (Equation (51)) allow us to define a generalization of Lah number. If we take $\mathbb{F}(U, Z) = U^r X^s$, then

$$M_{\bar{g}}(v\mathbb{F}) = M_{\bar{g}} \cdot (u\mathbb{E} \circ (\int Q_{m,\alpha})^r \cdot (Q_{m,\alpha})^s = M_{\bar{g}} \circ (uQ_{m,\alpha'})^r \cdot (Q_{m,\alpha})^s, \quad (58)$$

where $Q_{m,\alpha'}$ is the $\mathbb{L}$-specie of $m$-Stirling permutation weighted by $\alpha'(\sigma) = x^{mn}$ if $\sigma \in Q_m[n]$. Given that $Q_{m,\alpha}(t) = \frac{x}{(1-mx^m t)^{\frac{1}{m}}}$, $Q_{m,\alpha'}(t) = \frac{1}{(1-mx^m t)^{\frac{1}{m}}}$ and $M_{\bar{g}}(t) = v \exp(\frac{\mu u^m x^m t}{1-mx^m t})$, we have:

$$M_{\bar{g}}(v\mathbb{F})(t) = vu^r x^s \exp\left(\frac{\mu u^m x^m t}{1 - mx^m t}\right) (1 - mx^m t)^{-\frac{s+r}{m}}. \quad (59)$$

The equation (58) also shows that a $M_{\bar{g}}(v\mathbb{F})$-structure on $[n]$ is a triplet $s = (s_1, s_2, s_3)$, such that $s_1$ is a $M_{\bar{g}}$-structure on $A$, of weight $v\mu^k u^{mk} x^{mn_1}$, if $|A| = n_1$ and $s_1$ has $k$ connected components, $s_2$ is a $(uQm, \alpha')^r$-structure on $B \subset [n] \setminus A$, of weight $u^s x^{mn_2}$, where $n_2 = |B|$, and $s_3$ is a $(Qm, \alpha)^r$-structure of weight $x^{m(n-n_1-n_2)+r}$. The total weight of $s$ is therefore $v\mu^k u^{mk+s} x^{mn+r}$, if $s$ has $k$ connected components. Setting $M_{\bar{g}}^n(vu^s x^r) = vx^{mn+r} \sum_{k=1}^n L_{m,r,s}(n, k)\mu^k u^{mk+s}$, we have:

$$L_{m,r,s}(n+1, k) = L_{m,r,s}(n, k-1) + (m(n+k) + r + s)L_{m,r,s}(n, k). \quad (60)$$

Notice that the matrix $\Psi_{\mathbb{F}} = (L_{m,r,s}(n, k))_{n,k}$ of $M_{\bar{g}}(v\mathbb{F})$ satisfy (theorem 1) $\Psi_{\mathbb{F}} = C_{m,r}W_{m,s}$, where $C_{m,r} = (c_{m,r}(n, k))_{n,k}$ is the matrix of $\mathbb{H}_g(z\mathbb{F})$ (equation (24)), and $W_{m,r} = (W_{m,r}(n, k))_{n,k}$ is the matrix of $\mathbb{H}_{2,g}(z\mathbb{F})$. That is:

$$L_{m,r,s}(n, k) = \sum_{i=k}^{n} c_{m,r}(n, i) W_{m,s}(i, k). \quad (61)$$

Equations (26) and (42) show that:

$$L_{m,r,s}(n, k) = \sum_{i=k}^{n} \left( \sum_{j=i}^{n} \binom{j}{i} m^{n-j} r^{j-i} c(n, j) \right) \left( \sum_{l=0}^{i-k} \binom{i}{l} s^l m^{i-l-k} S(i-l, k) \right),$$
thus

$$L_{m,r,s}(n,k) = \sum_{k \le l \le i \le j \le n} \sum_{l=k}^{i} m^{n+i-j-k} r^{j-i} s^{i-l} c(n,j) S(l,k). \quad (62)$$

Equation (58) also allow us to write:

$$L_{m,r,s}(n,k) = \sum_{i=k}^{n} \binom{n}{i} L_m(i,k) \left( \sum_{j=0}^{n-i} \binom{n-i}{j} \left| (Q_m)^s[j] \right| \left| (Q_m)^r[n-i-j] \right| \right),$$

with $|(Q_m)^s[j]| = \sum_{l=0}^{j} m^{j-j} c(j,l) s^l |(Q_m)^r[n-i-j]| = \sum_{l'=0}^{n-i-j} m^{n-i-j-l'} c(n-i-j, l') r^{l'}$ (see (27)). Notice that the numbers $(L_{m,r,r}(n,k))$ is the r-Whitney-Lah numbers, defined by Eszter Gyimesi and Gábor Nyul in [6].

The grammar $H_m = \{z \to \mu z x^2, x \to m x^2\}$ from equation (57) allows us to define an other generalization of $(L_m)$ and of the Lah numbers. Indeed, by taking $\mathbb{F}(X) = X^r$, we have $\mathbb{H}_m(z\mathbb{F}) = \mathbb{H}_m \cdot (\mathbb{L}_{\gamma'})^r$, where $\mathbb{L}_{\gamma'}$ is the $\mathbb{L}$-species of lists weighted by $\gamma'(\sigma) = m^n x^{n+1}$ if $\sigma \in \mathbb{L}[n]$. That is,

$$\mathbb{H}_m(z\mathbb{F}) = (z\mathbb{E}_\beta \circ \mathbb{L}_\gamma^+) \cdot (\mathbb{L}_{\gamma'})^r, \quad (63)$$

the generating series being $\mathbb{H}_m(z\mathbb{F})(t) = z \exp(\frac{\mu x t}{1-mxt}) \frac{x^r}{(1-mxt)^r}$. Moreover, an $\mathbb{H}_m(z\mathbb{F})$-structure is a pair $s = (s_1, s_2)$, where $s_1$ is an $\mathbb{H}_m$-structure on $A$ weighted by $m^{|A|-k} \mu^k z x^{|A|+k}$ if it has $k$ connected components, and $s_2$ is a $(\mathbb{L}_{\gamma'})^r$-structure on $[n] \setminus A$, weighted by $m^{n-|A|} x^{n-|A|+r}$. The total weight of $s$ is $m^{n-k} \mu^k z x^{n+r}$, let us define $\mathcal{H}_m^n(zx^r) = \sum_{k=1}^r \tau_{m,r}(n,k) m^{n-k} \mu^k z x^{n+k+r}$. We have the recurrence:

$$\tau_{m,r}(n+1,k) = \tau_{m,r}(n,k-1) + m(n+k+r)\tau_{m,r}(n,k). \quad (64)$$

Note also that $\mathcal{H}_m^n(x^r) = r(r+1)\cdots(r+n-1)m^n x^{r+n}$. That is $|(\mathbb{L}_{\gamma'})^r[n]| = r(r+1)\cdots(r+n-1)m^n$. Thus:

$$\tau_{m,r}(n,k) = \sum_{i=k}^{n} \binom{n}{i} m^{n-k} L(i,k) r(r+1) \cdots (r+n-i-1). \quad (65)$$

We also have:

$$m^n(x+r)^{(n)} = \sum_{k=0}^{n} \tau_{m,r}(n,k) m^k x_{(k)} \quad (66)$$

Indeed, if we set $m^n(x+r)^{(n)} = \sum_{k=0}^{n} a(n,k) m^k x_{(k)}$, then $a(n+1,k) = a(n,k-1) + m(n+k+r)a(n,k)$.

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