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arXiv · 2606.00646

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

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

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Yao Dong, Chao Xu. 2026-05-30. A Refinement of the Fixed--Pixed Points Equidistribution on restricted Permutations. https://arxiv.org/abs/2606.00646

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