On the $\mathrm{rix}$ statistic and 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 $\mathrm{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 $\mathrm{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin and Zeng in the study of the $\mathrm{rix}$ 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 $\mathrm{fix}$ is homomesic under cyclic valley-hopping.

27 pages

References in corpus (4)

• Actions on permutations and unimodality of descent polynomials
• Gamma-positivity in combinatorics and geometry
• Homomesies on permutations -- an analysis of maps and statistics in the FindStat database
• Foatic actions of the symmetric group and fixed-point homomesy