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Historiographers theory and combinatorial
interpretations of Jacobi's elliptic functions

Benjamin Randrianirina

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Benjamin Randrianirina. Historiographers theory and combinatorial interpretations of Jacobi's elliptic functions. 2025. hal-05160123

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Open licence - etalab
# Historiographers theory and combinatorial interpretations of Jacobi's elliptic functions

Benjamin RANDRIANIRINA *

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

## Abstract

The concept of historiographer was introduced by G.X. Viennot to analyze the combinatorial solutions of combinatorial differential equations and, in particular, to study Jacobi's elliptic functions. We show how additive historiographers are related to system of differential equation. We apply this notion to hyperelliptic historiographer. In particular, we show that the notion of historiographer provides a link between various combinatorial interpretations of Jacobi's elliptic functions.

## 1 Introduction

Based on an unpublished manuscript by Professor Gérard Viennot, this article was written in 1998. It constitutes Chapter 3 of [12], written under the supervision of Pr Gilbert Labelle and Pr Pierre Leroux.

The notion of historiographer was introduced in the 1980s by Viennot in [15], to analyze the solutions of systems of combinatorial differential equations and in particular, to study Jacobi's elliptic functions. In this article, we introduce the notion of additive historiographers and study its connection between systems of differential equations (analytic or combinatorial), and William Chen's grammars (see [2]). Let us recall that, under suitable conditions, one can associate a William Chen grammar with a system of combinatorial differential equations, and conversely, any such system can be derived from a corresponding William Chen's grammar. More precisely, it was shown in [12] (theorem 10.5) or [13] (theorem 2.3.6) that:

**Theorem 1.1** Given a family of asymmetric B-species {$F_i(X_1, X_2, \dots, X_n), i = 1, \dots, n$}, where exponential generating series are {$F_i(x_1, x_2, \dots, x_n), i = 1, \dots, n$}, the following data are equivalent:

1. Providing the system of combinatorial differential equations:

$$ \{Y'_i = F_i(Y_1, \dots, Y_n), Y_i(0) = X_i, 1 \le i \le n\} \qquad (1) $$

<footnote>*Email: benjamin.randrianirina@univ-fianarantsoa.mg</footnote>
2. Providing the combinatorial differential operator:

$$ \mathcal{D} = \sum_{i=1}^{n} F_i(X_1, \dots, X_n) \frac{\partial}{\partial X_i} \qquad (2) $$

3. Providing the grammar G:

$$ G = \{x_i \to F_i(x_1, \dots, x_n), 1 \le i \le n\} \qquad (3) $$

4. Providing the system of analytical differential equations:

$$ \{y_i'(t) = F_i(y_1, \dots, y_n), 1 \le i \le n\} \qquad (4) $$

Solutions of (1) is the family of $\vec{F}$-enriched increasing rooted trees $\{Y_i = e^{\vec{T}}(X_i), i = 1 \cdots n\}$, and solutions of (4) satisfy:

$$ y_i(t) = T_{\vec{X}=\vec{x}}(e^{\vec{T}}(X_i))(t) = \mathrm{Gen}(x_i, t) \qquad (5) $$

where $T_{\vec{X}=\vec{x}}(e^{\vec{T}}(X_i))$ is the $\mathbb{L}$-species of isomorphism type of the mixed species $e^{\vec{T}}(X_i)$ and $\mathrm{Gen}(x_i, t) = \sum_{n \ge 0} \mathcal{G}^n(x_i) \frac{t^n}{n!}$, $\mathcal{G}$ being the differential operator associated with the grammar G.

Note that the $\mathbb{B}$-species $F(X_1, \dots, X_n) = \sum_{\vec{i}=(i_1, \dots, i_n)} a_{j, \vec{i}} X_1^{i_1} \cdots X_n^{i_n}$ is asymmetric and his exponential generating series is $F(x_1, \dots, x_n) = \sum_{\vec{i}=(i_1, \dots, i_n)} a_{j, \vec{i}} x_1^{i_1} \cdots x_n^{i_n}$.

For information on William Chen's grammar see [2] and [6]; on theory of species, see [1]; on mixed species, see [13] or [12] or [11]; and on systems of combinatorial differential equations, see [8], [9], and [10]. Initial conditions of the form $y(0) = x$, where $x$ is a constant or a formal variable, are studied in [12] and [13] or [11]. This condition correspond to $Y(0) = \mathbf{1}_x$, where $\mathbf{1}_x$ is the species of empty set weighted by $x$. The exponential generating series of $\mathbb{L}$-species of types $T_{\vec{X}=\vec{x}}(e^{\vec{T}}(X_i))$, $1 \le i \le n$ are solutions of the system of combinatorial differential equations $\{y'_i = F_i(y_1, \dots, y_n), y_i(0) = x_i, 1 \le i \le n\}$.

We apply the result to the hyperelliptic historiographers, which are the historiographers associated with the hyperelliptic species ([13] or [12] or [11]). We show, in particular, that various combinatorial interpretations of the Jacobi's elliptic functions are associated with the elliptic historiographer or Schett's historiographer. Therefore, on the one hand, we show that a combinatorial interpretation of these functions is associated with the way to build the combinatorial solutions of the system of combinatorial differential equations associated with the elliptic historiographer. On the other hand, this historiographer can be used to relate different combinatorial interpretations of the Jacobi's elliptic functions. Recall that the Jacobi's elliptic functions are the functions $\mathrm{sn}(t) = x(t)$, $\mathrm{cn}(t) = y(t)$, and $\mathrm{dn}(t) = z(t)$ solutions of the system:

$$ \left\{ \begin{array}{ll} x'(t) = y(t)z(t), & x(0) = 0 \\ y'(t) = -x(t)z(t), & y(0) = 1 \\ z'(t) = -k^2x(t)y(t), & z(0) = 1 \end{array} \right. \qquad (6) $$

Some realizations of Schett's historiographer were communicated to us by Viennot. ([17]).
## 2 Generalities about historiographers

**Definition 2.1 (cf [15])** An historiographer is a triplet $\mathcal{H} = (k, \mathcal{A}, p)$ where:

1. $k \in \mathbb{N}^*$

2. $\mathcal{A}$ is a finite set of mappings $A : \mathbb{N}^k \to \mathbb{N}^k$ called primitive operations.

3. $p : \mathcal{A} \times \mathbb{N}^k \to \mathbb{N}$ is a map, called possibility function.

The elements of $\mathbb{N}^k$ are called states, and the set $D(A) = \{u \in \mathbb{N}^k | p(A, u) \neq 0\}$ is the domain of the operation $A$.

The integer $p(A, u)$ is the number of ways to transition from state $u$ to state $v = A(u)$, using the operation $A$. Furthermore, these choices can be numbered from 1 to $p(A, u)$. It is therefore identified with the totally ordered set $P(A, u) = \{1, 2, \dots, p(A, u)\}$, which we call the set of possibilities. Note that if $u \notin D(A)$, then $p(A, u) = 0$ and $P(A, u) = \emptyset$.

Let $\mathcal{A}^*$ the set of all finite words over the alphabet $\mathcal{A}$ and $w = w_1w_2\cdots w_n \in \mathcal{A}^*$, $u_0 \in \mathbb{N}^k$. By defining $u_i = w_i(u_{i-1})$, we have a sequence $u = \{u_0, u_1, \dots, u_n\}$ of states $u_0$ is called the initial state and $u_n$ the final state of $u$. One can move from $u_0$ to $u_n$ by following the path $c(w) = u_0 \xrightarrow{w_1} u_1 \xrightarrow{w_2} u_2 \cdots \xrightarrow{w_n} u_n$. The number of ways to do this is $p(w) = \prod_{i=1}^n p(w_i, u_{i-1})$, and $p(u) \neq 0$ if and only if $u_{i-1} \in D(w_i)$ for all $i$.

**Definition 2.2 (cf [15])** An initialized historiographer is a pair $(\mathcal{H}, u_0)$, where $\mathcal{H}$ is a historiographer and $u_0$ is a fixed initial state. A history $h$ of size $|h| = n$ of $(\mathcal{H}, u_0)$ is a sequence $h = (w, m_1, m_2, \dots, m_n)$, where $w = w_1w_2\cdots w_n \in \mathcal{A}^*$ is a word such that $p(w) \neq 0$, and each $m_i$ is an integer such that for all $i$, $1 \leq m_i \leq p(w_i, u_{i-1})$.

In the following, we define $\mathcal{A} = \{A_1, A_2, \dots, A_r\}$. To each primitive operation $A_i$, we associate a formal variable $a_i$. Let $\vec{x} = \{x_1, x_2, \dots, x_k\}$ be a family of formal variables. We denote by $\langle \vec{x} \rangle^c = \langle x_1, \dots, x_k \rangle^c$ the set of commutative words in $\vec{x}$. Consider the application $\alpha : \mathbb{N}^k \to \langle \vec{x} \rangle^c$ such that $\alpha(n_1, n_2, \dots, n_k) = x_1^{n_1}x_2^{n_2}\cdots x_k^{n_k}$. If $w = w_1w_2\cdots w_n \in \mathcal{A}^*$, we denote $c_n(w) = a_1^{\mu_1}a_2^{\mu_2}\cdots a_r^{\mu_r}$, where $\mu_i = |\{j \in [n] | w_j = A_i\}|$.

Let $\mathcal{H}_{u_0}[n]$ the set of history of size $n$, associated with the initialized historiographer $(\mathcal{H}, u_0)$. For any $h = (w, m_1, m_2, \dots, m_n) \in \mathcal{H}_{u_0}[n]$, if $c(w) = u_0 \xrightarrow{w_1} u_1 \xrightarrow{w_2} u_2 \cdots \xrightarrow{w_n} u_n$, then the transition $u_{i-1} \xrightarrow{w_i} u_i$ was carried out by choosing $m_i$. We set $c_n(h) = c_n(w)$ and $u_n(h) = u_n$. The weight of $h$ is given by $\pi(h) = c_n(w)\alpha(u_n(h)) = a_1^{\mu_1}a_2^{\mu_2}\cdots a_r^{\mu_r}x_1^{n_1}x_2^{n_2}\cdots x_k^{n_k}$, if $u_n(h) = (n_1, n_2, \dots, n_k)$. We thus obtain the weighted $\mathbb{L}$-species $\mathcal{H}_{u_0}$, whose generating series is:

$$ \mathcal{H}_{u_0}(t) = \sum_{n \ge 0} |\mathcal{H}_{u_0}[n]|_{\pi} \frac{t^n}{n!} \quad (7) $$

$$ \text{where } |\mathcal{H}_{u_0}[n]|_{\pi} = \sum_{|h|=n} c_n(w) \alpha(u_n(h)), \quad |\mathcal{H}_{u_0}[0]| = \alpha(u_0). $$

**Definition 2.3 (cf [15])** A realization (or implementations) of an initialized historiographer $(\mathcal{H}, u_0)$ is a pair $(\mathbb{F}, \varphi)$, where $\mathbb{F}$ is a weighted $\mathbb{L}$-species and $\varphi = (\varphi_n)_{n \in \mathbb{N}}$ is a family of isomorphisms of weighted sets $\varphi_n : \mathbb{F}[n] \to \mathcal{H}_{u_0}[n]$.

The following theorem is obvious.

**Theorem 2.4** Two realizations of $(\mathcal{H}, u_0)$ are isomorphic.
### 3 Additive historiographer

**Definition 3.1** An historiographer $\mathcal{H} = (k, \mathcal{A}, p)$ is additive if:

1. For all $A \in \mathcal{A}$, the element $A(u) - u$ of $\mathbb{Z}^k$ does not depend on $u$; that is, there exists a vector $\vec{n}(A) \in \mathbb{Z}^k$, called the step, such that for all $u \in \mathbb{N}^k$, $A(u) = u + \vec{n}(A)$.

2. For all $u, v \in \mathbb{N}^k$, $A \in \mathcal{A}$, $p(A, u + v) = p(A, u) + p(A, v)$.

**Example 3.2** The historiographer $\mathcal{H}_m^{(k)} = (k, \mathcal{A}, p)$, where $m \ge 2$ and $\mathcal{A} = \{A_{1,i}, A_{2,i}, A_{3,i}, 1 \le i \le k\}$, with, for $i$:

* $A_{1,i}(a_1, \dots, a_i, \dots, a_k) = (a_1, \dots, a_{i-1}, a_i - 1, a_{i+1}, \dots, a_k)$,

* $A_{2,i}(a_1, \dots, a_i, \dots, a_k) = (a_1, \dots, a_{i-1}, a_i + m - 1, a_{i+1}, \dots, a_k)$

* $A_{3,i}(a_1, \dots, a_i, \dots, a_k) = (a_1 + m - 1, \dots, a_i + m - 1, a_i, a_{i+1} + m - 1, \dots, a_k + m - 1)$

and the function $p$ is defined by $p(A_{j,i}, u) = a_i$, if $u = (a_1, \dots, a_k)$, is additive: for $1 \le i \le k$, if $u = (a_1, \dots, a_k)$ then $\vec{n}(A_{1,i}, u) = (n_1, \dots, n_k)$, where $n_j = 0$ if $j \ne i$ et $n_i = -1$, $\vec{n}(A_{2,i}, u) = (n_1, \dots, n_k)$, where $n_j = 0$ if $j \ne i$ et $n_i = m - 1$, $\vec{n}(A_{3,i}, u) = (n_1, \dots, n_k)$, where $n_j = m - 1$ if $j \ne i$ et $n_i = 0$.

In the following, we will consider an additive historiographer $\mathcal{H} = (k, \mathcal{A}, p)$, and for all $i, 1 \le i \le k$, we note $e_i = (\delta_i^1, \dots, \delta_i^k)$, where $\delta_i^j = 0$ if $j \ne i$ and $\delta_i^i = 1$.

**Remark: 3.3** Knowing that for all $A \in \mathcal{A}$, $p_A = p(A, \cdot) : (\mathbb{N}^k, +) \to (\mathbb{N}, +)$ is a homomorphism of additive monoids, we have:

1. For all $A \in \mathcal{A}$, $p(A, \vec{0}) = 0$, that is $\vec{0} \notin D(A)$,

2. For every $A \in \mathcal{A}$, the map $p_A : \mathbb{N}^k \to \mathbb{N}$, defined by $p_A(u) = p(A, u)$, is entirely determined by the values $p_A(e_i)$ for $i \in [k]$:

$$p_A(a_1, \dots, a_k) = a_1 p_A(e_1) + \dots + a_k p_A(e_k).$$

**Lemma 3.4** If $u, u'$ and $u''$ are elements of $\mathbb{N}^k$ such that $u = u' + u''$, then $u \in D(A)$ if and only if $u' \in D(A)$ or $u'' \in D(A)$. In particular, for every $A \in \mathcal{A}$, if $D(A) \ne \emptyset$, then there exists at least one $i$ such that $e_i \in D(A)$.

*Proof:* $p_A(u) = p_A(u') + p_A(u'') > 0 \Rightarrow p_A(u') > 0$ or $p_A(u'') > 0$. If for all $i$ $p_A(e_i) = 0$ and if $u = (a_1, \dots, a_k)$ then $p_A(u) = a_1 p_A(e_1) + \dots + a_k p_A(e_k) = 0$. ◇

**Lemma 3.5** If $e_i \in D(A)$ and $\vec{n}(A) = (n_1(A), n_2(A), \dots, n_k(A))$, then for all $j \ne i$, $n_j(A) \ge 0$ and $n_i(A) \ge -1$. In particular, if there exist $i \ne j$ such that $e_i \in D(A)$ and $e_j \in D(A)$, then $n_s(A) \ge 0$ for all $s \in [k]$.

*Proof:* $A(e_i) = e_i + \vec{n}(A) = (n_1(A), \dots, n_{i-1}(A), n_i(A) + 1, n_{i+1}(A), \dots, n_k(A)) \in \mathbb{N}^k$. Thus, if $j \ne i$, $n_j(A) \ge 0$ and $n_i(A) \ge -1$. ◇

**Proposition 3.6** If $u_0 = u'_0 + u''_0$, then we have the isomorphism of weighted species $\mathcal{H}_{u_0} \sim \mathcal{H}_{u'_0} \cdot \mathcal{H}_{u''_0}$. In other words, the map $\varphi : \mathbb{N}^k \to \mathbb{L}_{\mathcal{E}_s}$ such that $\varphi(u) = \mathcal{H}_u$ is a morphism from the additive monoid $\mathbb{N}^k$ to the multiplicative monoid $\mathbb{L}_{\mathcal{E}_s}$ of weighted L-species.
**Proof:** By induction, we show that for every $n$, there exists an isomorphism of weighted sets $\varphi_n : \mathcal{H}_{u_0}[n] \to (\mathcal{H}_{u'_0} \cdot \mathcal{H}_{u''_0})[n]$.

If $n=0$, then $\mathcal{H}_{u_0} = \{u_0\}$ and $u_0$ can be identified to the quadruple $(\emptyset, \emptyset, u'_0, u''_0)$. Suppose that for every history $h = (w, m_1, m_2, \dots, m_n) \in \mathcal{H}_{u_0}[n]$, we can associate a unique quadruple $(l', l'', h', h'')$ such that $l' + l'' = [n]$, $h' = (w', m'_1, m'2, \dots, m'_s) \in \mathcal{H}_{u'_0}[l']$, $h'' = (w'', m''_1, m''2, \dots, m''_t) \in \mathcal{H}_{u''_0}[l'']$, with $s = |l'|$, $t = |l''|$, and whose final states satisfy $u_n = u'_s + u''_t$. Let $h = (w, m_1, m_2, \dots, m_n, m_{n+1})$ be a history whose path is $c(w) = u_0 \xrightarrow{w_1} u_1 \xrightarrow{w_2} u_2 \cdots \xrightarrow{w_s} u_n \xrightarrow{w_{n+1}} u_{n+1}$. Let $h_{[n]}$ denote the restriction of $h$ to $[n]$, that is, $h_{[n]} = (w_{[n]}, m_1, m_2, \dots, m_n)$, where $c(w_{[n]}) = u_0 \xrightarrow{w_1} u_1 \xrightarrow{w_2} u_2 \cdots \xrightarrow{w_s} u_n$. Then $h_{[n]}$ can be identified with a quadruple $(l'_{[n]}, l''_{[n]}, h'_{[n]}, h''_{[n]})$ where $l'_{[n]} + l''_{[n]} = [n]$, $h'_{[n]} = (w'_{[n]}, m'_1, m'_2, \dots, m'_s) \in \mathcal{H}_{u'_0}[l'_{[n]}]$, $h''_{[n]} = (w'_{[n]}, m''_1, m''_2, \dots, m''_t) \in \mathcal{H}_{u''_0}[l''_{[n]}]$ and $u'_s + u''_t = u_n$. Lemma 3.4 show that $u'_s \in D(w_{n+1})$ or $u''_t \in D(w_{n+1})$. On the other hand, we know that $1 \le m_{n+1} \le p(w_{n+1}, u_n) = p(w_{n+1}, u'_s) + p(w_{n+1}, u''_t)$. Let's define $(l', l'', h', h'')$ as follows:

1. If $u'_s \in D(w_{n+1})$ and $1 \le m_{n+1} \le p(w_{n+1}, u'_s)$, then we set $l' = l'_{[n]} \cup \{n+1\}$, $m'_{s+1} = m_{n+1}$, $w'_{s+1} = w_{n+1}$, $w' = w'_{[n]} w'_{s+1}$ and $h' = (w', m'_1, \dots, m'_s, m'_{s+1})$, $l'' = l''_{[n]}$, $h'' = h''_{[n]}$.

2. If $u'_s \in D(w_{n+1})$ and $p(w_{n+1}, u'_s) + 1 \le m_{n+1} \le p(w_{n+1}, u'_s) + p(w_{n+1}, u''_s)$, or if $u'_s \notin D(w_{n+1})$ then $p(w_{n+1}, u''_s) \ne 0$. We set $l'' = l''_{[n]} \cup \{n+1\}$, $m''_{s+1} = m_{n+1}$, $w''_{s+1} = w_{n+1}$, $w'' = w''_{[n]} w''_{s+1}$ and $h'' = (w'', m''_1, \dots, m''_s, m''_{s+1})$, $l' = l'_{[n]}$, $h' = h'_{[n]}$.

Assume that $u'_s \in D(w_{n+1})$ and $1 \le m_{n+1} \le p(w_{n+1}, u'_s)$. By setting $u'_{s+1} = w_{n+1}(u'_s)$, we have $u_{n+1} = u'_{s+1} + u''_t$. Indeed $w_{n+1}(u_n) - u_n = w_{n+1}(u'_s) - u'_s$ because $H$ is additive. Therefore, $u_{n+1} - u_n = w'_{s+1} - u'_s$, which implies that $u_{n+1} = w'_{s+1} + u''_s$. Hence $\pi(h) = \pi(h') \cdot \pi(h'')$. $\diamond$

**Corollary 3.7** If $u_0 = (\alpha_1, \alpha_2, \dots, \alpha_k) \in \mathbb{N}^k$, then we have the isomorphism

$$ \mathcal{H}_{u_0} \cong \Pi_{i=1}^{k} \mathcal{H}_{e_i}^{\alpha_i}. \qquad (8) $$

In other words, the set $\{\mathcal{H}_{u_0}, u_0 \in \mathbb{N}^k\}$ is completely determined by the data of the set $\{\mathcal{H}_{e_1}, \mathcal{H}_{e_2}, \dots, \mathcal{H}_{e_k}\}$

Recall that each operator $A$ is associated with a variable $a(A)$, and $\mathcal{H}$ is additive, therefore if $e_i \in D(A)$ then $A(e_i) = e_i + \vec{n}(A) = (n_1(A), \dots, n_i(A) + 1, \dots, n_k(A))$.

**Proposition 3.8** Let $\mathcal{H} = (k, \mathcal{A}, p)$ an additive historiographer. We set $\mathcal{A}(i)$ the set of primitive operation $A \in \mathcal{A}$ such that $e_i \in D(A)$ and $X_i(t) = \mathcal{H}_{e_i}(t) = \sum_{n \ge 0} |\mathcal{H}_{e_i}[n]|_{\pi} \frac{t^n}{n!}$. Then the family $(X_i(t))_{1 \le i \le k}$ is a solution of the system of differential equations

$$ \frac{\partial}{\partial t} X_i(t) = \sum_{A \in \mathcal{A}(i)} a(A) p(A, e_i) X_1^{n_1(A)}(t) \cdots X_i^{n_i(A)+1}(t) \cdots X_k^{n_k(A)}, \qquad (9) $$

with $X_i(0) = x_i$, $1 \le i \le k$.

**Proof:** We first observe that a history $h$ of length zero is a history such that the word $w$ is empty. In this case, $h$ is reduced to $e_i$. Thus $\mathcal{H}_{e_i}[0] = \{e_i\}$ and $|\mathcal{H}_{e_i}|_{\pi} = \alpha(e_i) = x_i$.

A history $h \in \mathcal{H}_{e_i}[n+1]$ is completely determined by a triple $(A, m_1, h')$ such that $A \in \mathcal{A}(i)$, $1 \le m_1 \le p(A, e_i)$, and $h' \in \mathcal{H}_{A(e_i)}[n]$. Indeed, let $h = (w, m_1, \dots, m_n, m_{n+1}) \in$
$\mathcal{H}_{e_i}[n+1]$. Clearly, $w_1 \in \mathcal{A}(i)$. We define $u_1 = w_1(e_i) = (n_1(A), \dots, n_i(A)+1, \dots, n_k(A))$, and $h' = (w', m'_1, \dots, m'_n)$ with $w = w'w'$ and $m'_i = m_{i+1}$. Notice that we have $\pi(h) = a(w_1)\pi(h')$. Conversely, if $u_1 \in \mathbb{N}^k$ and $A \in \mathcal{A}$, such that $u_1 = A(e_1)$, and $h' = (w, m'_1, \dots, m'_n) \in \mathcal{H}_{u_1}[n]$, and if $1 \le m_1 \le p(A, e_i)$, then $h = (w, m_1, m'_1, \dots, m'_n) \in \mathcal{H}_{e_i}[n+1]$.

We can deduce that:

$$
\begin{align*}
\frac{\partial}{\partial t} \mathcal{H}_{e_i}(t) &= \sum_{n \ge 0} \left[ \sum_{h \in \mathcal{H}_{e_i}[n+1]} \pi(h) \right] \frac{t^n}{n!} \\
&= \sum_{n \ge 0} \left[ \sum_{A \in \mathcal{A}(i)} \sum_{1 \le m \le p(A, e_i)} \sum_{h \in \mathcal{H}_{A(e_i)}} a(A) \pi(h) \right] \frac{t^n}{n!} \\
&= \sum_{A \in \mathcal{A}(i)} a(A) \sum_{1 \le m \le p(A, e_i)} \sum_{n \ge 0} \sum_{h \in \mathcal{H}_{A(e_i)}} \pi(h) \frac{t^n}{n!} \\
&= \sum_{A \in \mathcal{A}(i)} a(A) \sum_{1 \le m \le p(A, e_i)} \mathcal{H}_{A(e_i)}(t) \\
&= \sum_{A \in \mathcal{A}(i)} a(A) p(A, e_i) \mathcal{H}_{A(e_i)}(t)
\end{align*}
$$

Equation (8) say that $\mathcal{H}_{A(e_i)} \simeq \mathcal{H}_{e_1}^{n_1(A)} \cdots \mathcal{H}_{e_i}^{n_i(A)+1} \cdots \mathcal{H}_{e_k}^{n_k(A)}$. ♦

We obtain the following theorem:

**Theorem 3.9** Let $(a_j)_j$ be formal variables and $(n_{ij})$, $(\alpha_{ij})$ be non negative integers, where $1 \le i \le k$, $1 \le j \le r$. Then, the following data are equivalent:

1. Providing the additive historiographer

$$
\mathcal{H} = (k, \mathcal{A}, p), \tag{10}
$$

$$
\text{with } \mathcal{A} = \{A_j, 1 \le j \le r\}, A_j(e_i) = (n_{1,j}, \dots, n_{i,j}, \dots, n_{k,j}) \text{ and } p(A_j, e_i) = \alpha_{i,j}.
$$

2. Providing the system of analytical differential equation:

$$
\frac{\partial y_i}{\partial t}(t) = \sum_{j=1}^{r} \alpha_{ij} a_j y_1^{n_{1,j}}(t) \cdots y_i^{n_{i,j+1}}(t) \cdots y_k^{n_{k,j}}(t), \quad y_i(0) = x_i, \quad 1 \le i \le k, \quad (11)
$$

3. Providing the system of combinatorial differential equation:

$$
\frac{\partial Y_i}{\partial T} = \sum_{j=1}^{r} \alpha_{ij} a_j Y_1^{n_{1,j}} \cdots Y_i^{n_{i,j+1}} \cdots Y_k^{n_{k,j}}, \quad Y_i(0) = X_i, \quad 1 \le i \le k, \quad (12)
$$

4. Providing the grammar:

$$
G = \{x_i \to \sum_{j=1}^{r} \alpha_{ij} a_j x_1^{n_{1,j}} \cdots x_i^{n_{i,j+1}} \cdots x_k^{n_{k,j}}, \quad 1 \le i \le k\}. \quad (13)
$$

For $1 \le i \le k$, the $\mathbb{L}$-species $Y_i(T) = T_{\vec{X}=\vec{x}}(e^{T\mathcal{D}}(X_i))$ of type of $Y(T, \vec{X})$-structure, solution of (12), is a realization of initialized historiographer $\mathcal{H}_{e_i}$. Its exponential generating series is the solution of (11) and satisfy:

$$
y_i(t) = \mathcal{H}_{e_i}(t) = \mathrm{Gen}(x_i, t),
$$

where $\mathrm{Gen}(x_i, t) = \sum_{i \ge 0} \mathcal{G}^n(x_i) \frac{t^n}{n!}$, and $\mathcal{G} = \sum_{i=1}^k \sum_{j=1}^r \alpha_{ij} a_j x_1^{n_{1,j}} \cdots x_i^{n_{i,j+1}} \cdots x_k^{n_{k,j}} \frac{\partial}{\partial x_i}$ the differential operator associated with $G$.
<figure><img src="image_3.png" /><figcaption>Figure 1: Integral process associated with(12)</figcaption></figure>

*Proof:* Theorem 1.1 establishes the equivalence between 11, 12, and 13. Proposition 3.8 shows that the exponential generating series of the family {$\mathcal{H}_{e_1}, \mathcal{H}_{e_2}, \dots, \mathcal{H}_{e_k}$} are also solution of (11). It suffices to show that $3 \Rightarrow 1$. We associate to the system of combinatorial differential equations (12), the historiographer $H = (k, A, p)$ defined by $A = \{A_j, 1 \le j \le r\}$ with $A_j(e_i) = (n_{1,j}, \dots, n_{i,j}, \dots, n_{k,j})$ and $p(A_j, e_i) = \alpha_{ij}$. If $u = (u_1, u_2, \dots, u_n)$ then $p(A_j, u) = \sum_{i=1}^n \alpha_{ij} u_i$. We observe that the canonical solution of (12) is the family of increasing enriched trees $\mathcal{A} = (\mathbb{A}_i, 1 \le i \le k)$, constructed by iterating the integral process illustrated in Figure 1, and the exponential generating series are solution of (12). Thus, for all $i$, $\mathcal{H}_{e_i}(t) = \mathbb{A}_i(t)$. These $\mathbb{L}$-species are monomially weighted and reduced in the sense of Definition 3.8 in [12] (or [13] Definition 1.4.7). Theorem 3.9 of [12] (or [13] Theorem 1.4.8) then states that $\mathcal{H}_{e_i} = \mathbb{A}_i$. ◦

# 4 Applications to Hyperelliptic Historiographers

In this section, we analyze the so-called hyperelliptic historiographers. In particular, various realizations of Schett’s historiographers are studied.

## 4.1 Generality

**Definition 4.1** A hyperelliptic historiographer A hyperelliptic historiographer of order $k$, with $k \ge 2$, is a historiographer $\mathcal{H}^{(k)} = (k, A, p)$, where $A = \{A_1, A_2, \dots, A_k\}$, with $A_i(\alpha_1, \dots, \alpha_k) = (\alpha_1 + 1, \dots, \alpha_{i-1} + 1, \alpha_i - 1, \alpha_{i+1} + 1, \dots, \alpha_k + 1)$, and $p(A_i, u) = \alpha_i$ if $u = (\alpha_1, \dots, \alpha_i, \dots, \alpha_k)$.

The historiographer $\mathcal{H}^{(k)}$ is Schett’s historiographer.

The following lemma is obvious.

**Lemma 4.2** The historiographer $\mathcal{H}^{(k)}$ is additive: for $1 \le i \le k$, if $u = (a_1, \dots, a_k)$ then $\vec{n}(A_i, u) = (n_1, \dots, n_k)$, where $n_j = 1$ if $j \ne i$ end $n_i = 0$.

According to the theorem 3.9, the system of analytical differential equation associated with $\mathcal{H}^{(k)}$ is:

$$ \frac{\partial y_i}{\partial t}(t) = a_i y_1(t) \cdots y_{i-1}(t) y_{i+1}(t) \cdots y_k(t), \quad y_i(0) = x_i, \quad 1 \le i \le k, \qquad (14) $$

the system of combinatorial differential equation is:

$$ \frac{\partial Y_i}{\partial T} = a_i Y_1 \cdots Y_{i-1} Y_{i+1} \cdots Y_k, \quad Y_i(0) = X_i, \quad 1 \le i \le k, \qquad (15) $$

and the grammar is:

$$ G^{(k)} = \{x_i \to a_i x_1 \cdots x_{i-1} x_{i+1} \cdots x_k^{n_{k,j}}, \quad 1 \le i \le k\}. \qquad (16) $$
We denote by $\mathcal{D}^{(k)}$ the combinatorial differential operator associated to (15). Its solution is the family of mixed species $\{F(i\vec{X}) = e^{\mathrm{TD}^{(k)}}(X_i)\}, 1 \le i \le k\}$. Theorem 3.9 says that for $1 \le i \le k$, $\mathcal{H}_{e_i}^{(k)}$ is isomorphic to the $\mathbb{L}$-species $T_{\vec{X}=\vec{x}} e^{\mathrm{TD}^{(k)}}(X_i)$. A history $h \in \mathcal{H}^{(k)}[n]$ such that $\pi(h) = \Pi_{j=1}^k a_j^{\mu_j} x_j^{\alpha_j}$, where $\mu_j = |\{s|\omega_s = A_j\}$ and the final states is $u = (\alpha_1, \dots, \alpha_k)$, can be identified with a tree $a$ having $\mu_j$ vertices colored by $a_j$ and $n_j$ leaves of type $j$.

<figure><img src="image_4.png" /><figcaption>Figure 2: A tree weighted by $a_1 a_2 a_3^2 x_1^3 x_2^2$</figcaption></figure>

**Example 4.3** Let us consider Schett’s historiographer $\mathcal{H}^{(3)}$. Figure 2 represents an element of $(T_{\vec{X}=\vec{x}} e^{\mathrm{TD}^{(3)}}(X_1))[4]$ with weight $a_1 a_2 a_3^2 x_1^3 x_2^2$. It corresponds to the history $h = A_1 A_2 A_3 A_3 1121 \in \mathcal{H}_{e_1}^{(3)}[4]$, whose final state is $u_4 = (3, 2, 0)$. Figure 3 shows the successive step in the construction of this tree. The final state associated with each step is respectively $u_1 = (0, 1, 1)$, $u_2 = (1, 0, 2)$, $u_3 = (2, 1, 1)$ and $u_4$.

<figure><img src="image_5.png" /><figcaption>Figure 3: The successive steps in construction of the tree in Figure 2</figcaption></figure>

Note that the exponential generating series $\mathcal{H}_{e_i}^{(k)}(t) = \mathrm{Gen}(x_i, t)$, is the solution of the equation $\frac{\partial y_i}{\partial t} = a_i y_1 \cdots y_{i-1} y_{i+1} \cdots y_k$, $y_i(0) = x_i$. Hence $|\mathcal{H}_{e_i}^{(k)}[n]|_\pi = (\mathcal{G}^{(k)})^n(x_i)$.

**Definition 4.4** For $1 \le i \le k$, the polynomials $H_n^{(k,i)}(a_1, \dots, a_k, x_1, \dots, x_k) = (\mathcal{G}^{(k)})^n(x_i)$, $n \in \mathbb{N}$ is called the hyperelliptic polynomial of order $k$, and $H_n^{(3,i)}(a_1, \dots, a_k, x_1, \dots, x_k)$, $n \in \mathbb{N}$ is Schett’s polynomials (see Dumont [5]).

These polynomials satisfy the recurrence relation $H_0^{(k,i)}(a_1, \dots, a_k, x_1, \dots, x_k) = x_i$, and

$$H_{n+1}^{(k,i)} = \sum_{i=1}^{k} a_i x_1 \cdots x_{i-1} x_{i+1} \cdots x_n \frac{\partial H_n^{(k,i)}}{\partial x_i} \quad (17)$$

We have the following lemma, which generalizes the theorem 1 of [14].

**Lemma 4.5** Let $h = (w_1, \dots, w_n, m_1, \dots, m_n) \in \mathcal{H}_{e_i}^{(k)}[n]$ an history whose final state is $u = (\alpha_1, \dots, \alpha_n)$ and the weight is $\pi(h) = \Pi_{j=1}^k a_j^{\mu_j} x_j^{\alpha_j}$, where $\mu_j = |\{s|w_s = A_j\}|$. Then the integers $\mu_j$ and $\alpha_j$ satisfy the relations:

$$ \begin{align} (i) \sum_{i=1}^{k} \mu_j &= n \tag{ii} \\ (ii) \sum_{s=1}^{k} \alpha_s &= n(k-2)+1 \tag{18} \\ (iii) \mu_i + \alpha_i &= 1 + \sum_{s=1, s \neq i}^{k} \mu_s \tag{iv} \\ (iv) \text{ if } j \neq i, \text{ then } \mu_j + \alpha_j &= \sum_{s=1, s \neq j}^{k} \mu_s \notag \end{align} $$
*Proof:* Relation (i) is obvious.

We prove (ii) by induction on $n$. Suppose that for $n' < n$ and for every $h' \in \mathcal{H}_{e_i}^{(k)}[n']$ such that the weight is $\pi(h') = \Pi_{j=1}^k a_j^{\mu'_j} x_j^{\alpha'_j}$, we have $\sum_{s=1}^k \alpha'_s = n'(k-2)+1$. Let $h = (w_1, \cdots, w_n, m_1, \cdots, m_n) \in \mathcal{H}_{e_i}^{(k)}[n]$ an history whose final state is $u = (\alpha_1, \cdots, \alpha_n)$. We may assume that $i=1$. That is $w_1 = A_1 = 1$ and $m_1 = 1$. Let $c(h) = e_1 \xrightarrow{A_1} u_1 \xrightarrow{w_2} u_2 \cdots \xrightarrow{w_0} u_n$ the associated path, and $h' = w_2 \cdots w_n m_2 \cdots m_n \in \mathcal{H}_{u_1}^{(k)}[n]$. It is clear that the final state of $h'$ is $u_n$, and its weight is $\pi(h') = a_1^{\mu_1-1} x_1^{\alpha_1} \Pi_{j=2}^k a_j^{\mu'_j} x_j^{\alpha'_j}$. We have $u_1 = (0,1,\cdots,1)$. Proposition 3.6 say that $\mathcal{H}_{u_1} \simeq \mathcal{H}_{e_2} \cdots \mathcal{H}_{e_k}$. Then $h'$ can be identified with $(l_2, \cdots, l_k, h_2, \cdots, h_k)$, where $l_1+_o \cdots+_o l_k = [n-1]$, and for $i=2, \cdots, k$ $h_i \in \mathcal{H}_{e_i}^{(k)}(l_i)$. Suppose that $\pi(h_i) = \Pi_{j=1}^k a_j^{\mu_{ij}} x_j^{\alpha_{ij}}$. Since $\pi(h') = \pi(h_2) \cdots \pi(h_k)$, we have $\alpha_j = \sum_{i=2}^k \alpha_{i,j}$, for all $j$. By the induction hypothesis, we have $\sum_{j=1}^k \alpha_{ij} = |l_i|(k-2)+1 = n(k-2)+1$. Hence $\sum_{i=1}^k \alpha_i = \sum_{i=1}^k \sum_{i=2}^k \alpha_{i,j} = \sum_{i=1}^k |l_i|(k-2)+1 = n(k-2)+1$.

<figure><img src="image_6.png" /><figcaption>Figure 4: Bijective proof for (iii)</figcaption></figure>

Figure 4, where $k=4$ and $i=1$, presents a bijective proof for (iii) and Figure 5, with $j=2$, presents a bijective proof for (iv).

<figure><img src="image_7.png" /><figcaption>Figure 5: Bijective proof for (iv)</figcaption></figure>

Note that a proof of (iii) and (iv) by induction are possible. ◇

**Corollary 4.6** Let $h = (w_1, \cdots, w_n, m_1, \cdots, m_n) \in \mathcal{H}_{e_j}^{(k)}$ such that the final state is $u_n = (\alpha_1, \cdots, \alpha_{i-1}, 0, \alpha_{i+1}, \cdots, \alpha_k)$. If $j=i$ then $n$ is odd and $\mu_i = \frac{n+1}{2}$. If $i \neq j$, then $n$ is even and $\mu_i = \frac{n}{2}$.

*Proof:* If $i=j$, then the relations (i) and (iii) of (18) show that $2\mu_i = n+1$. And if $i \neq j$, the relation (iv) show that $\mu_i = \sum_{s \neq i} \mu_s$. Knowing that $\mu_i + \sum_{s \neq i} \mu_s = n$ (relation (i)), we have $2\mu_i = n$. ◇

It follows that if $u_n = (\alpha_1, \cdots, \alpha_n)$ is a final state such that $\alpha_i = 0$, then for all $j \neq i$, $\alpha_j \neq 0$.
## 4.2 Realizations of Schett's historiographer and Jacobi elliptic functions

The purpose of this paragraph is to show that various combinatorial interpretations of the analytic solutions of a system of differential equations can arise from the way in which the historiographer associated with the system is realized. We examine the case of Jacobi's elliptic functions, which are essentially the solutions of the system associated with Schett's historiographer $S = \mathcal{H}^{(3)} = (3, A, p)$ where $A = \{A, B, C\}$, with $A(u) = (\alpha_1 - 1, \alpha_2 + 1, \alpha_3 + 1)$, $B(u) = (\alpha_1 + 1, \alpha_2 - 1, \alpha_3 + 1)$, $C(u) = (\alpha_1 + 1, \alpha_2 + 1, \alpha_3 - 1)$, $p(A, u) = \alpha_1$, $p(B, u) = \alpha_2$ and $p(C, u) = \alpha_3$ if $u = (\alpha_1, \alpha_2, \alpha_3)$ under the condition $x_1 = x = 0$. Let us consider the system associated with Schett's historiographer $S$:

$$ \begin{cases} X' = aYZ, & X(0) = x \\ Y' = bYZ, & Y(0) = y \\ Z' = cXZ, & Z(0) = x \end{cases} \qquad (19) $$

Let us recall that Schett's polynomials (definition 4.4) are $X_n(a, b, c, x, y, z) = n![t^n]X(t)$, $Y_n(a, b, c, x, y, z) = n![t^n]Y(t)$ and $Z_n(a, b, c, x, y, z) = n![t^n]Z(t)$, where $X(t) = X(t; a, b, c, x, y, z)$, $Y(t) = Y(t; a, b, c, x, y, z)$ and $Z(t) = Z(t; a, b, c, x, y, z)$ are the analytical solutions of (19). We propose five different realizations of Schett's historiographer. This yields five different interpretations of Schett's polynomials. Using Remark 4.7, we thus obtain five different combinatorial interpretations of the Jacobi's elliptic functions. Note that in [7], Flajolet and Françon propose a combinatorial interpretation of Jacobi's elliptic functions in terms of doubled permutations, using continued fractions. This interpretation does not fit within the framework of historiographers.

**Remark: 4.7** 1. Taking $x = 0$ is equivalent to considering the final states $u = (0, \alpha_2, \alpha_3)$. Let $h \in S_{e_i}[n]$ such that $\pi(h) = a^{\mu_{i1}}b^{\mu_{i2}}c^{\mu_{i3}}x^{\alpha_{i1}}y^{\alpha_{i2}}z^{\alpha_{i3}}$, such that $\alpha_{i1} = 0$. Corollary 4.6 says that if $i = 1$ then $n$ is odd and if $i = 2$ or $i = 3$, then $n$ is even. That is the function $X(t) = \sum_{n \ge 0} X_n(a, b, c, 0, y, z) \frac{t^n}{n!}$ is odd, functions $Y(t) = \sum_{n \ge 0} Y_n(a, b, c, 0, y, z) \frac{t^n}{n!}$ and $Z(t) = \sum_{n \ge 0} Z_n(a, b, c, 0, y, z) \frac{t^n}{n!}$ are even. Note also that for all $n \in \mathbb{N}$, $Y_n(a, b, c, 0, 1, 1) = Z_n(a, b, c, 0, 1, 1).$

2. Taking $a = 1, b = -1, c = -k^2$, $x = 0, y = z = 1$, we have $\operatorname{sn}(t) = X(t)$, $\operatorname{cn}(t) = Y(t)$. In other words, $a_{2n+1,p} = (-1)^n(2n+1)![k^{2p}t^{2n+1}]\operatorname{sn}(t) = (2n+1)![a^{n+1}b^{n-p}c^p]X(t; a, b, c, 0, 1, 1)$, and $a_{2n} = (-1)^n(2n)![k^{2p}t^{2n}]\operatorname{cn}(t) = (2n)![a^nb^{n-p}c^p]Y(t; a, b, c, 0, 1, 1).$

3. Setting $X(t) = \sum_{n \ge 0}(\sum a_{i_1,i_2,i_3}a^{i_1}b^{i_2}c^{i_3}x^{j_1}y^{j_2}z^{j_3})\frac{t^n}{n!}$, $Y(t) = \sum_{n \ge 0}(\sum b_{h_1,h_2,h_3}a^{h_1}b^{h_2}c^{h_3}x^{s_1}y^{s_2}z^{s_3})\frac{t^n}{n!}$, and $Z(t) = \sum_{n \ge 0}(\sum a_{r_1,r_2,r_3}a^{r_1}b^{r_2}c^r x^{q_1}y^{q_2}z^{q_3})\frac{t^n}{n!}$, Schett shows that ([14] theorem II) $a_{i_1,i_2,i_3} = b_{i_3,i_1,i_2} = c_{i_2,i_3,i_1}$, $a_{i_1,i_2,i_3} = a_{i_1,i_3,i_2}$, $b_{i_3,i_1,i_2} = b_{i_2,i_1,i_3}$ and $c_{i_2,i_3,i_1} = c_{i_3,i_2,i_1}$.

4. In [16], Viennot shows that $\sum_p a_{2n+1,p} = e_{2n+1}$ and $\sum_p a_{2n,p} = e_{2n}$, where $e_n$ is the Euler's numbers or tangent and secant numbers (See also Dumont [4]).

A story $h = (w_1, \dots, w_n, m_1, \dots, m_n)$ will be represented by an increasing binary tree, solution to the system (19), whose leaves are weighted by $x, y$, or $z$ and the internal vertices are weighted by $a, b$, or $c$. In accordance with the methods of Leroux and Viennot for solving equations and systems of differential equations (see [9]), this tree is
constructed recursively as follows. Suppose that we have constructed the tree associated with the history $h_i = (w_1 \cdots w_i, m_1, \cdots, m_i)$, whose final state is $u_i = (\alpha_i, \beta_i, \gamma_i)$. We have $p(A, u_i) = \alpha_i$, $p(B, u_i) = \beta_i$ and $p(C, u_i) = \gamma_i$. Each leaf of type $x$ is numbered from 1 to $\alpha_i$, each leaf of type $y$ from 1 to $\beta_i$, and each leaf of type $z$ from 1 to $\gamma_i$, from left to right. Suppose that $w_{i+1} = A$. Then we replace the leaf of type $x$, numbered $m_i$ with a node labeled $i+1$ and weighted by $a$, followed by two new leaves, of type $y$ and $z$ respectively. In the following, we will consider, in all cases, as an example of realizations, that of the history $h_0 = (ACABBABCACAAA; 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1)$.

### 4.2.1 First canonical realization

In this realization ([17]), when vertex $i+1$ is inserted on a leaf $x$, $y$ is placed to the left of $z$; when it is inserted on $y$, $x$ is placed to the left of $z$; and when it is inserted on $z$, $x$ is also placed to the left of $y$, as shown in Figure 6 (A). In Figure 6 (B), we have the

<figure><img src="image_8.png" /><figcaption>Figure 6: (A): Insertion of a vertex $i$; (B): First canonical realization of $h_0$</figcaption></figure>

increasing binary tree corresponding to $h_0$. Note that this realization corresponds to the noncommutative grammar $G = \{x \to yaz, ay \to xbz, z \to xcy\}$.

Observe that in an increasing binary tree, the path from a leaf to the root is an alternation of sequences of left edges and right edges. Let $s$ be a left (respectively, right) leaf such that the path from $s$ to 1 consists of the sequence $c_1 = (s, s_1, s_2, \cdots, s_k)$ of left (respectively, right) edges, followed by the sequence $c_2 = (s_k, s_{k+1}, \cdots, s_{k+r})$ of right (respectively, left) edges, and then the sequence $c_3 = (s_{k+r}, \cdots, s_{k+r+t}), \cdots$. Then $c_1$ is called the left (respectively, right) branch of $s$, with root $s_k$. The length $r$ of $c_2$ is called the arm of $s$ and is denoted by $\beta(s)$ or $\beta(c_1)$. For example, in Figure 6 (B), the left branch of the leaf $y$ associated with 12 is $(y, 12, 7, 3, 2)$, and its arm is $\beta(y) = 1$. We call the left branch (respectively, right branch) with $\beta(g) = 0$ the principal left branch (respectively, principal right branch).

Let $g$ be a left branch of a tree $b$, which is the realization of a history $h \in S_{e_1}(n)$ (respectively $h \in S_{e_2}(n)$):

1. If $\beta(g)$ is even and $g$, then the root of $g$ is weighted by $b$, and if $\beta(g)$ is odd, then its root is weighted by $c$.

2. If the final state of $h$ is $u = (0, \alpha_2, \alpha_3)$ ($x=0$) and $h \in S_{e_1}(n)$ then $n$ is odd, the principal left branch of $b$ has an odd length; the others have even lengths. And if $h \in S_{e_2}(n)$ then $n$ is even, and all left branches have even length.

An increasing binary tree on $[n]$ in which all left branches have even length, except the principal left branch when $n$ is odd, is called a Jacobi tree. For example, the tree on Figure 6 (B) is a Jacobi tree. Corollary 4.6 and remark 4.7 shows that:
**Theorem 4.8** *The coefficient $a_{2n+1,p} = (-1)^n(2n+1)![k^{2p}t^{2n+1}]\text{sn}(t)$ (respectively, $a_{2n,p} = (-1)^n(2n)![k^{2p}t^{2n}]\text{cn}(t)$) is equal to the number of Jacobi's trees on $[2n+1]$ (respectively, on $[2n]$) such that the number of left branches $g$ for which $\beta(g)$ is odd is equal to $p$.*

### 4.2.2 Second canonical realization

We consider the realization corresponding to the noncommutative grammar $G = \{x \rightarrow yaz, y \rightarrow zbx, z \rightarrow xcy\}$. In this case, if we insert $i+1$ on $y$, then $x$ is placed to the left of $z$, the rest is left unchanged, as shown in Figure 7 (A). Figure 7 represents the realization

<figure><img src="image_9.png" /><figcaption>Figure 7: (A): Insertion of a vertex $i$; (B): Second canonical realization of $h_0$</figcaption></figure>

of $h_0$. We call the left height (respectively, right height) of a vertexs the number of left edges (respectively, right edges) on the path from $s$ to the root 1. We denote $h_l(s)$ the left height of $s$, and $h_r(s)$ its right height.

**Lemma 4.9** Let $\tau(s)$ be the second canonical realization of a history $h \in S_{e_1}[n]$ (respectively $h \in S_{e_2}[n]$). Then:

1. The vertex $s$ is weighted by $a$, or $s$ is a leaf of type $x$ if and only if $h_r(s) - h_l(s) \equiv 0 \pmod{3}$ (respectively, $h_r(s) - h_l(s) \equiv 1 \pmod{3}$).

2. The vertex $s$ is weighted by $b$, or $s$ is a leaf of type $y$ if and only if $h_r(s) - h_l(s) \equiv 2 \pmod{3}$ (respectively, $h_r(s) - h_l(s) \equiv 0 \pmod{3}$).

3. The vertex $s$ is weighted by $c$, or $s$ is a leaf of type $z$ if and only if $h_r(s) - h_l(s) \equiv 1 \pmod{3}$ (respectively, $h_r(s) - h_l(s) \equiv 2 \pmod{3}$).

*Proof:* By induction on $n$. Let $h$ be a history on $[n+1]$, and let $h$ be the history on $[n]$ obtained by forgetting $w_{n+1}$. It is clear that $\tau(h)$ is obtained by removing $n+1$ from $\tau(h')$, and conversely, $w_{n+1}$ allows us to insert $n+1$ into $\tau(h)$ to obtain $\tau(h')$. If the vertex $n+1$ in $\tau(h')$ is weighted by $a$, then $w_{n+1} = A$ and $n+1$ has replaced a leaf of type $x$. In this case, $h_r(n+1) = h_r(x)$ and $h_l(n+1) = h_l(x)$. Thus $h_r(n+1) - h_l(n+1) = h_r(x) - h_l(x) \equiv 0 \pmod{3}$. Moreover, a new left leaf $y$ and a new right leaf $z$ appear. We have $h_l(y) = h_l(x) + 1$, $h_r(y) = h_r(x)$, $h_l(z) = h_l(x)$, and $h_r(z) = h_r(x) + 1$. Then $h_r(y) - h_l(y) = h_r(x) - h_l(x) - 1 \equiv 2 \pmod{3}$, et $h_r(z) - h_l(z) = h_r(x) - h_l(x) + 1 \equiv 1 \pmod{3}$. Conversely, let $c = (1, s_1, \dots, s)$ be a path in a tree associated with a history $h \in S_{e_1}[n]$ such that $h_r(s) - h_l(s) \equiv 0 \pmod{3}$. If the edge $(1, s_1)$ is a left edge, then $c$ can be identified with the path $c_1 = (s_1, \dots, s)$, in a tree associated with a history $h' \in [k]$ in which $h_r(s) - h_l(s) \equiv 1 \pmod{3}$. And if the edge $(1, s_1)$ is a right edge, then $c$ can be identified with the path $c_1 = (s_1, \dots, s)$, in a tree associated with a history $h' \in S_{e_3}[k]$ in which $h_r(s) - h_l(s) \equiv 2 \pmod{3}$. Therefore, $s$ is either a leaf of type $x$ or a vertex
weighted by $a$. We use the same reasoning in the case where $n + 1$ is weighted by $b$ or $c$. ◇

We obtain the following proposition:

**Proposition 4.10** Let $\mathcal{B}_{2n+1}$ be the set of increasing binary trees on $[2n+1]$ such that, for every vertex $s$: if $h_r(s) - h_l(s) \equiv 1 \pmod 3$, then $s$ has a left child; and if $h_r(s) - h_l(s) \equiv 2 \pmod 3$, then $s$ has a right child. And let $\mathcal{B}_{2n}$ be the set of increasing binary trees on $[2n]$ such that, for every vertex $s$: if $h_r(s) - h_l(s) \equiv 2 \pmod 3$, then $s$ has a left child; and if $h_r(s) - h_l(s) \equiv 0 \pmod 3$, then $s$ has a right child. Then:

1. The coefficient $a_{2n+1,p} = (-1)^n(2n+1)![k^{2p}t^{2n+1}]\text{sn}(t)$ is equal to the number of elements in $\mathcal{B}_{2n+1}$ having exactly $p$ vertices $s$ such that $h_r(s) - h_l(s) \equiv 1 \pmod 3$.

2. The coefficient of $a_{2n,p} = (-1)^n(2n)![k^{2p}t^{2n}]\text{cn}(t)$ is equal to the number of elements in $\mathcal{B}_{2n}$ having exactly $p$ vertices $s$ such that $h_r(s) - h_l(s) \equiv 2 \pmod 3$.

3. For all $n \in \mathbb{N}$, $|\mathcal{B}_n| = e_n$.

*Proof:* Lemma 4.9 shows that the elements of $\mathcal{B}_{2n+1}$ (respectively, $\mathcal{B}_{2n}$) are the realizations of the elements of $S_{e_1}[2n+1]$ (respectively, $S_{e_1}[2n]$) with no leaves of type $x$. ◇

### 4.2.3 Viennot's realization

Viennot's realization ([15]) is a variant of the two previous realizations, following the scheme described in Figure 8 (I). That is if $h_r(x)$ is odd, the leaf $y$ is placed to the right of $z$ and if $h_r(x)$ is even, $y$ is placed to the left of $z$. Figure 8 (II) is a realization of $h_0$. By considering trees that do not contain any leaf of type $x$, we obtain a Jacobi trees. It

<figure><img src="image_10.png" /><figcaption>Figure 8: (I): Insertion of $i$; (II) Viennot's realization of $h_0$</figcaption></figure>

is clear that a Jacobi tree corresponds to a Jacobi permutation in the sense of Definition 3 of [16]. The following theorem is then a reformulation of theorem 1 of [16].

**Theorem 4.11** The coefficient $a_{2n+1,p} = (-1)^n(2n+1)![k^{2p}t^{2n+1}]\text{sn}(t)$ (respectively, $a_{2n,p} = (-1)^n(2n)![k^{2p}t^{2n}]\text{cn}(t)$) is equal to the number of Jacobi's trees on $[2n+1]$ (respectively, on $[2n]$) such that the number of vertices with odd right height is $p$.

### 4.2.4 Alternating realization

In this paragraph, we will construct a realization ([17]) in such a way that, by taking $x = 0$ and $y = z = 1$, we obtain an interpretation of Jacobi elliptic functions in terms of complete increasing binary trees, and hence of alternating permutations. Figure 9 illustrates the conditions under which an alternating realization is possible. That is:
<figure><img src="image_11.png" /><figcaption>Figure 9: Condition for an alternating realization</figcaption></figure>

1. No leaf of type $y$ or $z$ can appear to the left or right of a leaf of type $x$, unless $w_1 = B$ or $C$, and $h_r(y) = 1$ or $h_r(z) = 1$.

2. A leaf of type $y$ (respectively, type $z$) is accompanied on its right (respectively, its left) by a leaf $y$ or by a leaf $z$.

<figure><img src="image_12.png" /><figcaption>Figure 10: Insertion of $i$</figcaption></figure>

An element $i$ is inserted according to the rules illustrated in Figure 10. Figure 11 present the alternating realization of $h_0$.

<figure><img src="image_13.png" /><figcaption>Figure 11: Alternating realization of $h_0$</figcaption></figure>

### 4.2.5 Dumont's realization

Let us recall that $i$ is a cycle peak on $\sigma \in S_n$ if $\sigma^{-1}(i) < i > \sigma(i)$. We denote by $i(\sigma)$ (respectively, $p(\sigma)$) the number of edges incident to a odd cycle peak (respectively, even cycle peak) in $\sigma$, and $b(\sigma)$ the number of edges that are not incident to any cycle peak. In [3], Dumont shows that the polynomials $\tilde{X}_n(x, y, z) = X_n(1, 1, 1, x, y, z)$, satisfy:

$$ \left\{ \begin{array}{l} \tilde{X}_{2n}(x, y, z) = x \sum_{\sigma \in S_{2n}} x^{i(\sigma)} y^{p(\sigma)} z^{b(\sigma)} \\ \\ \tilde{X}_{2n+1}(x, y, z) = y \sum_{\sigma \in S_{2n+1}} x^{i(\sigma)} y^{p(\sigma)} z^{b(\sigma)} \end{array} \right. \qquad (20) $$
Note that in the representation of a permutation by an increasing binary tree, the cycle peaks correspond to the leaves, except for the leaf with right height 0, which corresponds to a fixed point if it exists. Thus, the increasing binary tree in Figure 12 corresponds

<figure><img src="image_14.png" /><figcaption>Figure 12: The increasing binary tree associated with $\sigma' = (7)(5, 8, 6)(4)(1, 3, 2, 9)$</figcaption></figure>

to the permutation $\sigma = 758641329$, which can be identified with the permutation $\sigma' = (7)(5, 8, 6)(4)(1, 3, 2, 9)$ written in cycle notation. The cycle peak of $\sigma'$ are 8, 3 and 9. Then, for a Dumont's realization, the following conditions must be satisfied (Figure 13):

* Every cycle peak labeled with an even number (respectively, an odd number) is followed by two unlabeled leaves weighted by $x$ (respectively, by $y$).

* An unlabeled leaf associated with a labeled vertex that is not a cycle peak is weighted by $z$.

* The unlabeled leaf of right height 0 is weighted by $x$ if $n$ is even, and by $y$ if $n$ is odd.

<figure><img src="image_15.png" /><figcaption>Figure 13: Condition for an Dumont's realization</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14: Insertion of an element $i$</figcaption></figure>
So, when an integer $i$ is inserted, then the weight of the leaf of right height 0 varies according to the parity of $i$. Moreover, if $i$ is even, then the unlabeled leaves associated with $ii$ are weighted by $x$, and if $i$ is odd, they are weighted by $y$. Then, element $i$ is inserted following the rule illustrated in the figure 14.

<figure><img src="image_17.png" /><figcaption>Figure 15: Dumont's realization of $h_0$</figcaption></figure>

Figure 15 shows Dumont’s realization of $h_0$.

## 4.3 Bijection between different combinatorial interpretations of Schett’s elliptic functions.

Recall that an isomorphism of weighted $L$-species is a weight-preserving bijection (see [13]). According Theorem 2.4, the different realizations of Schett’s historiographers, are isomorphic. They satisfy the system of differential equation (19). In the same way, various combinatorial interpretations of Schett’s elliptic functions, obtained by specializing $x = 0$ in these realizations, are also isomorphic. In this subsection, we illustrate, through two examples, how to construct this isomorphism. This construction proceeds as follows:

1. Start with a given structure in the first model;

2. Reconstruct the history of this structure;

3. Build the realization of this history in the second model.

**Example 4.12** In this example we shows how to construct a bijection between Viennot’s model and Dumont’s model. Let us consider the Jacobi permutation $\sigma = 587214963$. Assuming that $\sigma \in S_{e_1}[9]$, we can verify that it is a realization of the history $h = (ACBAACBAA, 1, 1, 2, 2, 1, 2, 1, 1, 1)$. The weight of $\sigma$ is $a^5b^2c^2y^5z^5$. It suffices then to construct the Dumont’s model associated with $h$.

Figure 16 represents this construction. We obtain the permutation $\sigma' = (9)(6)(5, 8, 7)$ (1, 2, 3, 4), which contains no odd cycle peaks and two even cycle peaks.

**Example 4.13** In the following example we illustrates the bijection between alternating permutations (alternating realizations) and Jacobi permutations (Viennot realizations). Let us consider the alternating permutation $\sigma = 745281936$. Assuming that $\sigma \in S_{e_1}[9]$, we can reconstruct its history. We obtain $h = (ABACCCAAA, 1, 1, 1, 1, 1, 1, 1, 1)$. On the other hand, the weight of $\sigma$ is $a^5bc^3y^7z^3$.

The Viennot’s realization associated with $h$ gives us the Jacobi permutation $\sigma' = 761853296$, which is the image of $\sigma$ by this isomorphism.
<figure><img src="image_18.png" /><figcaption>Figure 16: Isomorphism between Viennot's model and Dumont's model</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 17: Isomorphism between Viennot's model and alternating model</figcaption></figure>

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