Partial $\gamma$-Positivity for Quasi-Stirling Permutations of Multisets

TL;DR

This paper proves that certain polynomials counting quasi-Stirling permutations of multisets are partial gamma-positive, confirming a recent conjecture and linking these permutations to ordered labeled trees.

Contribution

It establishes the partial gamma-positivity of polynomials for quasi-Stirling permutations and related trees, providing new combinatorial proofs and confirming conjectures.

Findings

Proves partial gamma-positivity for quasi-Stirling permutations

Links permutations to ordered labeled trees for combinatorial proofs

Provides alternative proof for Stirling permutations' gamma-positivity

Abstract

We prove that the enumerative polynomials of quasi-Stirling permutations of multisets with respect to the statistics of plateaux, descents and ascents are partial $\gamma$-positive, thereby confirming a recent conjecture posed by Lin, Ma and Zhang. This is accomplished by proving the partial $\gamma$-positivity of the enumerative polynomials of certain ordered labeled trees, which are in bijection with quasi-Stirling permutations of multisets. As an application, we provide an alternative proof of the partial $\gamma$-positivity of the enumerative polynomials on Stirling permutations of multisets.