Nadia Lafrenière ; Yan Zhuang - On the $\operatorname{rix}$ statistic and valley-hopping

dmtcs:11553 - Discrete Mathematics & Theoretical Computer Science, March 5, 2024, vol. 26:2 -
On the $\operatorname{rix}$ statistic and valley-hoppingArticle

Authors: Nadia Lafrenière ; Yan Zhuang

    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.

    Volume: vol. 26:2
    Section: Combinatorics
    Published on: March 5, 2024
    Accepted on: January 5, 2024
    Submitted on: July 7, 2023
    Keywords: Mathematics - Combinatorics,05A05 (Primary), 05E18 (Secondary)

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