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

Nadia Lafrenière, Yan Zhuang

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 $γ$-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 $Φ$ 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.

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

Abstract

This paper studies the relationship between the modified FoataStrehl action (a.k.a. valley-hopping)a group action on permutations used to demonstrate the -positivity of the Eulerian polynomialsand the number of rixed points 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 statistic is homomesic under valley-hopping. We also demonstrate that a bijection introduced by Lin and Zeng in the study of the 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 is homomesic under cyclic valley-hopping.
Paper Structure (13 sections, 17 theorems, 44 equations, 3 figures, 1 algorithm)

This paper contains 13 sections, 17 theorems, 44 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Permutation statistics
  3. An iterative algorithm for rixed points and the rix-factorization
  4. Explicit description of the algorithm
  5. Proof that Algorithm \ref{['algorithm']} gives the rix-factorization
  6. Proof that Algorithm \ref{['algorithm']} gives the rixed points
  7. A characterization of rixed points
  8. Properties of rixed points and the rix-factorization
  9. Valley-hopping and cyclic valley-hopping
  10. rix is homomesic under valley-hopping
  11. Phi sends valley-hopping orbits to cyclic valley-hopping orbits
  12. The bijection Phi
  13. Proof of Theorem \ref{['t-main']}

Key Result

Lemma 9

At any stage during the execution of Algorithm algorithm, if the valid factor is $\pi_l\cdots\pi_r$, then $\pi_r, \pi_{r+1}, \ldots, \pi_n$ are all ascents of $\pi$, so $\pi_r< \pi_{r+1} < \cdots < \pi_n$.

Figures (3)

  • Figure 3.1: Visual depictions of Algorithm \ref{['algorithm']} on $\pi = 142785369$ and $\pi = 23816457$, as in Examples \ref{['eg-rix1-2']} and \ref{['eg-rix2-2']}. The peaks that mark the end of each $\alpha$-factor in the rix-factorization are depicted as green squares, and the rixed points as red diamonds. The progression of the valid factor is depicted on top of the permutation.
  • Figure 4.1: Valley-hopping on $\pi=834279156$ with $S=\{6,7,8\}$ yields $\varphi_{S}(\pi)=734289615$.
  • Figure 4.2: Cyclic valley-hopping on $\pi=(523)(8)(97641)=935124687$ with $S=\{3,7,8\}$ yields $\psi_{S}(\pi)=(8532)(96417)=782134956$.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Example 3
  • Example 4
  • Remark 6
  • Example 7: Example \ref{['eg-rix1']} continued
  • Example 8: Example \ref{['eg-rix2']} continued
  • Lemma 9
  • proof
  • Proposition 10
  • ...and 36 more