Combinatorics on bi-${\hat{I}}^3$-positivity of $1/k$-Eulerian polynomials

Abstract

The $1/k$-Eulerian polynomials $A_n^{(k)}(x)$ were introduced as ascent polynomials over $k$-inversion sequences by Savage and Viswanathan. The bi-${\hat{I}}^3$-positivity of the $1/k$-Eulerian polynomials $A_n^{(k)}(x)$ was known but to give a combinatorial interpretation of the corresponding bi-${\hat{I}}^3$-coefficients still remains open. The study of the theme of bi-${\hat{I}}^3$-positivities from purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-${\hat{I}}^3$-coefficients of $A_n^{(k)}(x)$ 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 ${\hat{I}}^3$-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.

30 pages, 11 figures