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.