On the $\operatorname{rix}$ statistic and valley-hopping

TL;DR

This paper explores the relationship between the rixed points statistic and valley-hopping group actions on permutations, providing algorithms and proofs of homomesy properties, and connecting these concepts to Eulerian polynomials.

Contribution

It introduces a linear-time algorithm for rixed points, proves homomesy of the rix statistic under valley-hopping, and links orbit structures via a bijection to cyclic valley-hopping.

Findings

Rixed points can be computed in linear time.

The rix statistic is homomesic under valley-hopping.

Fixed points are homomesic under cyclic valley-hopping.

Abstract

This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $\gamma$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping.