Homomesy in products of two chains

Abstract

International audience Many cyclic actions $τ$ on a finite set $\mathcal{S}$ ; of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit ``homomesy'': the average of $f$ over each $τ$-orbit in $\mathcal{S} $ is the same as the average of $f$ over the whole set $\mathcal{S} $. This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind and discuss old and new results for the actions of promotion and rowmotion on the poset that is the product of two chains.

The birational Lalanne-Kreweras involution

“…More precisely, if ϕ is an invertible operator acting on a combinatorial set X, and f : X → R is some statistic on X, then we say that f is homomesic with respect to the action of ϕ on X if the average of f along every ϕ-orbit is equal to the same constant. The study of homomesies for combinatorial operators was initiated by Propp and Roby [29].…”

“…The systematic investigation of homomesies was initiated by Propp and Roby [29]. There has been a particular emphasis on exhibiting homomesies for rowmotion, including its piecewise-linear and birational extensions [25,1,29,13,32,10,22,15,19,23].…”

Symmetry of Narayana numbers and rowvacuation of root posets

“…Indeed, the past 10 or so years has seen the emergence of the subfield of Dynamical Algebraic Combinatorics [22,24], in which rowmotion features prominently. Furthermore, Panyushev's "constant average cardinality along orbits" observation was one of the first instances of homomesy [20], a phenomenon that again is at the heart of Dynamical Algebraic Combinatorics.…”

Order polynomial product formulas and poset dynamics

“…Rotation of binary words, and parabolic cosets of Weyl groups. Under the "Stanley-Thomas word" bijection (see [65,54]), rowmotion of J (R(a, b)) corresponds to rotation of binary words with a 1's and b 0's. Extending this description, Rush and Shi [60] showed that if P is the minuscule poset corresponding to the minuscule weight λ, then under the natural isomorphism J (P ) ≃ W/W J , where W J is the parabolic subgroup of the Weyl group W stabilizing λ, the action of rowmotion is conjugate to the action of a Coxeter element c ∈ W .…”