A symmetry on weakly increasing trees and multiset Schett polynomials

TL;DR

This paper introduces a new symmetry in weakly increasing trees related to multiset Schett polynomials, providing combinatorial and algebraic proofs, and demonstrating properties like symmetry, unimodality, and partial gamma-positivity.

Contribution

It extends Schett polynomials to multisets, establishes a new symmetry in weakly increasing trees, and proves partial gamma-positivity, unifying combinatorial structures and elliptic functions.

Findings

A new combinatorial interpretation for Jacobi elliptic functions coefficients.

A proven symmetry in weakly increasing trees extending Schett polynomials.

Partial gamma-positivity of multiset Schett polynomials.

Abstract

By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma-Mansour-Wang-Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin-Ma-Ma-Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of "even-degree nodes on odd levels" and "odd-degree nodes" on weakly increasing trees is found, extending the Schett polynomials, a generalization of the Jacobi elliptic functions introduced by Schett, to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences. Moreover, via introducing a group action on trees, we prove the partial $\gamma$-positivity of the multiset Schett polynomials, a result implies both the symmetry and the unimodality of these polynomials.