The Eulerian Distribution on the Involutions of the Hyperoctahedral Group is Indeed $\gamma$-Positive

Abstract

Let $I_n^B$ denote the set of the involutions of the hyperoctahedral group $B_n$, and let $\mathrm{des}_B(\pi )$ denote the number of descents of the permutation $\pi \in B_n$. We settle a problem of Moustakas which states that

$$\begin{aligned} I_n^B(t):=\sum _{\pi \in I_n^B}t^{\mathrm{des}_B(\pi )} \end{aligned}$$

is $\gamma$-positive for $n\ge 1$.