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A combinatorial proof of Jacobi's elliptic identity via alternating permutations

Jean-christophe Pain

TL;DR

The paper addresses the problem of giving a direct combinatorial interpretation of Jacobi's differential identity in terms of elliptically weighted alternating permutations. It develops a unified framework linking Entringer numbers, Dumont–Viennot snakes, and elliptic continued fractions, where the derivative of the elliptic generating function $sn$ corresponds to a canonical factorization into $cn$ and $dn$ components. The main contributions include the canonical factorization $sn'(u)=cn(u)\,dn(u)$, a growth-operator viewpoint on differentiation, a snake-based interpretation, and elliptic J-fractions that align with Entringer refinements. This work bridges combinatorics and elliptic function theory, offering structural insights and potential generalizations.

Abstract

We provide a unified combinatorial framework connecting Entringer numbers, Dumont-Viennot snakes, and elliptically weighted continued fractions, which gives a structural interpretation of the Jacobi elliptic identity \begin{equation} \mathrm{sn}'(u)=\mathrm{cn}(u)\,\mathrm{dn}(u), \end{equation} where $\mathrm{sn}$, $\mathrm{cn}$ and $\mathrm{dn}$ are the Jacobi elliptic functions. This framework allows the decomposition of weighted snakes corresponding to the derivative of $\mathrm{sn}$ into canonical $\mathrm{cn}$- and $\mathrm{dn}$-components, bridging classical combinatorics and elliptic function theory.

A combinatorial proof of Jacobi's elliptic identity via alternating permutations

TL;DR

The paper addresses the problem of giving a direct combinatorial interpretation of Jacobi's differential identity in terms of elliptically weighted alternating permutations. It develops a unified framework linking Entringer numbers, Dumont–Viennot snakes, and elliptic continued fractions, where the derivative of the elliptic generating function corresponds to a canonical factorization into and components. The main contributions include the canonical factorization , a growth-operator viewpoint on differentiation, a snake-based interpretation, and elliptic J-fractions that align with Entringer refinements. This work bridges combinatorics and elliptic function theory, offering structural insights and potential generalizations.

Abstract

We provide a unified combinatorial framework connecting Entringer numbers, Dumont-Viennot snakes, and elliptically weighted continued fractions, which gives a structural interpretation of the Jacobi elliptic identity \begin{equation} \mathrm{sn}'(u)=\mathrm{cn}(u)\,\mathrm{dn}(u), \end{equation} where , and are the Jacobi elliptic functions. This framework allows the decomposition of weighted snakes corresponding to the derivative of into canonical - and -components, bridging classical combinatorics and elliptic function theory.
Paper Structure (20 sections, 1 theorem, 86 equations, 3 figures, 1 table)

This paper contains 20 sections, 1 theorem, 86 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation.
  3. Alternating permutations and Entringer numbers
  4. Elliptic weighting, local statistics and derivation as a growth operator
  5. Canonical factorization and proof of Jacobi's identity
  6. Connection with Dumont--Viennot snakes
  7. Jacobi continued fractions and Entringer refinements
  8. Recursive structure and differential relations
  9. Jacobi continued fraction for $\mathrm{sn}(u,k)$
  10. Analytic Jacobi fractions and comparison with the combinatorial model
  11. Combinatorial interpretation of coefficients
  12. Factorization under differentiation
  13. Conclusion
  14. The number of ascending alternating permutations and André's theorem
  15. Snakes and elliptic weights
  16. ...and 5 more sections

Key Result

Proposition 4.1

By the locality assumption on the elliptic statistic (section sec3), the weight is multiplicative under removal of the maximal element. Let $\mathcal{S}_{2n+1}$ denote the set of alternating permutations satisfying Let $\mathcal{S}_{2n+1}^\bullet$ be the corresponding class where one marks the maximal element. Then removal of the maximal element induces a weight-preserving bijection where $\mat

Figures (3)

  • Figure 1: A simple Dumont--Viennot snake corresponding to an alternating permutation. The elliptic node (shaded) contributes to a factor $k$ to the weight.
  • Figure 2: Removal of the maximal element splits a snake into two independent alternating sub-snakes.
  • Figure :

Theorems & Definitions (4)

  • Proposition 4.1: Canonical factorization of elliptic snakes
  • proof
  • Example 4.1
  • Definition B.1