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 $Π$, let $\mathfrak{S}_n(Π)$ denote the set of permutations in the symmetric group $\mathfrak{S}_n$ that avoid every element of $Π$ in the sense of pattern avoidance. For each set $Π$ appearing in their conjecture, we show that the pairs of statistics $(\mathrm{des},\mathrm{fix})$ and $(\mathrm{ides},\mathrm{pix})$ are equidistributed over $\mathfrak{S}_n(Π)$. Our proof is based on explicit ordinary generating functions for the corresponding pattern-avoiding classes.