Combinatorics on bi-$γ$-positivity of $1/k$-Eulerian polynomials

TL;DR

The paper resolves the open problem of a combinatorial interpretation for the bi-$\gamma$-coefficients of the $1/k$-Eulerian polynomials $A^{(k)}_n(x)$. It develops a forest-based model using increasing pruned even $k$-ary forests and establishes three main steps: bijections between $k$-Stirling permutations and these forests, a generalized Foata–Strehl action yielding $\gamma$-positivity for longest ascent-plateau polynomials with initial letter 1, and two key forest-transformations that produce explicit bi-$\gamma$-coefficients. The main result expresses the symmetric decomposition of $A^{(k)}_n(x)$ in terms of counts of forests with given numbers of old leaves and singletons, producing explicit $\gamma$-expansions with coefficients $\overline{\gamma}_{n,k,i}$ and $\widehat{\gamma}_{n,k,i}$. This combinatorial framework extends $\gamma$-positivity ideas to bi-$\gamma$-positivity for general $k$, and introduces forest-based mechanisms that may have independent interest for ascent statistics and related permutation classes.

Abstract

The $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ were introduced as ascent polynomials over $k$-inversion sequences by Savage and Viswanathan. The bi-$γ$-positivity of the $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ was known but to give a combinatorial interpretation of the corresponding bi-$γ$-coefficients still remains open. The study of the theme of bi-$γ$-positivities from purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-$γ$-coefficients of $A^{(k)}_{n}(x)$ by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps: (i) construct a bijection between $k$-Stirling permutations and certain forests that are named increasing pruned even $k$-ary forests; (ii) introduce a generalized Foata--Strehl action on increasing pruned even $k$-ary trees which implies the longest ascent-plateau polynomials over $k$-Stirling permutations with initial letter $1$ are $γ$-positive, a result that may have independent interest; (iii) develop two crucial transformations on increasing pruned even $k$-ary forests to conclude our combinatorial interpretation.

Figures (9)

• Figure 1: A forest $F\in \mathcal{F}_{10}(3)$.
• Figure 2: All $9$ forests in $\overline{\mathcal{F}}_{3}(2)$.
• Figure 3: All $6$ forests of $\widehat{\mathcal{F}}_{3}(2)$.
• Figure 4: An increasing pruned even $3$-ary forest $(T_1, T_2, T_3)$.
• Figure 5: The transformation $\Phi_{4}$.

Theorems & Definitions (25)

• Proposition 1.1
• Proposition 1.2
• Definition 2.1: Increasing pruned even $k$-ary tree
• Definition 2.2: Increasing pruned even $k$-ary forest
• Definition 2.3: Removable old leaf
• Theorem 2.4
• Example 2.5
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3