Homomesy on permutations with toggling actions

TL;DR

The paper investigates when permutation statistics have the same average across all orbits under the rotation action on $S_n$, uncovering 34 statistics that exhibit homomesy. By recasting rotation as toggling via adjacent transpositions and identifying right-multiplication by an $n$-cycle, the authors show these homomesies hold for a broader class of maps, linking to rowmotion and Coxeter elements in reflection groups. They also extend the analysis to restricted and modified toggles—Pair swapping, Parity rotation, and Valley hopping—deriving additional, parity-dependent homomesies. The work combines FindStat-guided candidate identification with rigorous orbit-structure proofs, revealing deep connections between permutation statistics, toggling dynamics, and representation-theoretic perspectives across groups. This framework advances understanding of invariance phenomena in dynamical algebraic combinatorics and suggests practical methods to identify homomesic statistics in related combinatorial actions.

Abstract

Homomesy is an invariance phenomenon in dynamical algebraic combinatorics which occurs when the average value of some statistic on a set of combinatorial objects is the same over each orbit generated by a map on these objects. In this paper we perform a systematic search for statistics homomesic for the set of permutations under the rotation map, identifying and proving 34 instances of homomesy. We show that these homomesies actually hold not only for rotation but in fact for a whole class of maps related to rotation by the notion of toggling, which is identified initially with composition of simple transpositions. In this way these maps are related to the rowmotion action defined on various combinatorial structures, which has a useful definition in terms of toggling. We prove some initial results on maps given by restricted or modified toggles. We discuss also the computational method used to identify candidate statistics from FindStat, a combinatorial statistics database.

Figures (4)

• Figure 1: Excedances, Deficiencies, and Fixed Points
• Figure 2: The $rot$ map represented as a composition of toggles.
• Figure 3: The $ps$ map represented as a composition of toggles.
• Figure 4: Valley Hopping on $\pi=246135$.

Theorems & Definitions (61)

• Remark 1.1
• Lemma 1.2
• Definition 1.3
• Example 1.4
• Lemma 1.5
• proof
• Lemma 1.6
• proof
• Theorem 2.1
• proof