Homomesy in products of two chains. (English) Zbl 1319.05151

Propp, James; Roby, Tom

Electron. J. Comb. 22, No. 3, Research Paper P3.4, 29 p. (2015).

Summary: 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 D. I. Panyushev [Eur. J. Comb. 30, No. 2, 586–594 (2009; Zbl 1165.06001)] in the context of the rowmotion action on the set of antichains of a root poset; D. Armstrong et al. [Trans. Am. Math. Soc. 365, No. 8, 4121–4151 (2013; Zbl 1271.05011)] proved Panyushev’s conjecture. 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.

MSC:

05E18	Group actions on combinatorial structures
05E10	Combinatorial aspects of representation theory
06A11	Algebraic aspects of posets

Keywords:

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