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

TL;DR

This paper provides a combinatorial interpretation for the bi-$ ext{γ}$-coefficients of $1/k$-Eulerian polynomials using ordered labeled forests, advancing understanding of their bi-$ ext{γ}$-positivity.

Contribution

It introduces a novel combinatorial approach involving forests and bijections to interpret bi-$ ext{γ}$-coefficients of $1/k$-Eulerian polynomials.

Findings

Established a bijection between $k$-Stirling permutations and certain forests.

Proved $ ext{γ}$-positivity of longest ascent-plateau polynomials over $k$-Stirling permutations.

Provided a combinatorial interpretation for bi-$ ext{γ}$-coefficients.

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-$\gamma$-positivity of the $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ was known but to give a combinatorial interpretation of the corresponding bi-$\gamma$-coefficients still remains open. The study of the theme of bi-$\gamma$-positivities from purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-$\gamma$-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 $\gamma$-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.