The Eulerian distribution on involutions is indeed $\gamma$-positive

Abstract

Let ${\mathcal{I}}_n$ and ${\mathcal{J}}_n$ denote the set of involutions and fixed-point free involutions of $\{1,\dots ,n\}$, respectively, and let $\text{des}(\pi )$ denote the number of descents of the permutation $\pi$. We prove a conjecture of Guo and Zeng which states that $I_n(t):={\sum }_{\pi \in {\mathcal{I}}_n}{t}^{\text{des}(\pi )}$ is $\gamma$-positive for $n\ge 1$ and $J_{2n}(t):={\sum }_{\pi \in {\mathcal{J}}_{2n}}{t}^{\text{des}(\pi )}$ is $\gamma$-positive for $n\ge 9$. We also prove that the number of $(3412,3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable permutations with $m$ double descents and $k$ descents.