Homomesy in products of two chains

James Propp¹ ; Tom Roby²

¹University of Massachusetts Lowell Department of Mathematics
²University of Connecticut Department of Mathematics

The electronic journal of combinatorics, Tome 22 (2015) no. 3

Résumé

Many invertible actions $\tau$ on a set $\mathcal{S}$ of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit the following property which we dub homomesy: the average of $f$ over each $\tau$-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 the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suter's action on certain subposets of Young's Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.

Détail

DOI : 10.37236/3579

Classification : 05E18, 05E10, 06A11
Mots-clés : antichains, Ballot theorems, homomesy, Lyness 5-cycle, orbit, order ideals, Panyushev complementation, permutations, poset, product of chains, promotion, rowmotion, sandpile, Suter's symmetry, toggle group, Young's lattice, Young tableaux