Homomesy in products of two chains

• Published in: Discrete Mathematics and Theoretical Computer Science Vol. DMTCS Proceedings vol. AS,...; no. Proceedings; pp. 945 - 956
• Main Authors: Propp, James, Roby, Tom
• Format: Journal Article Conference Proceeding
• Language: English
• Published: DMTCS 01.01.2013; Discrete Mathematics and Theoretical Computer Science; Discrete Mathematics & Theoretical Computer Science
• Series: DMTCS Proceedings
• Subjects: [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]; antichains; combinatorial ergodicity; Computer Science; Discrete Mathematics; homomesy; orbit; order ideals; poset; product of chains; promotion; rowmotion; sandpile; toggle group; rowmotion; product of chains; sandpile; combinatorial ergodicity; order ideals; homomesy; orbit; antichains; poset; toggle group; promotion
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.2356

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.