A Refinement of the Fixed--Pixed Points Equidistribution on restricted Permutations

Abstract

Motivated by a recent conjecture of Bsila, Cox, Hugo, Styron and Zhuang concerning fixed points and pixed points on pattern-avoiding permutations, we prove a bivariate refinement involving descent statistics. Given a set of permutations $\hat{I}$, let ${\mathfrak{S}}_n(\hat{I})$ denote the set of permutations in the symmetric group ${\mathfrak{S}}_n$ that avoid every element of $\hat{I}$ in the sense of pattern avoidance. For each set $\hat{I}$ appearing in their conjecture, we show that the pairs of statistics $(\mathrm{des},\mathrm{fix})$ and $(\mathrm{ides},\mathrm{pix})$ are equidistributed over ${\mathfrak{S}}_n(\hat{I})$. Our proof is based on explicit ordinary generating functions for the corresponding pattern-avoiding classes.