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arXiv · 2501.12055

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

Sherry H. F. Yan·Xubo Yang·Zhicong Lin
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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.

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Sherry H. F. Yan, Xubo Yang, Zhicong Lin. 2025-01-21. Combinatorics on bi-$\gamma$-positivity of $1/k$-Eulerian polynomials. https://arxiv.org/abs/2501.12055

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