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  1. Home > Articles & Issues >
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  3. Homomesy in products ...
Conference paper

Homomesy in products of two chains

James Propp (1), Tom Roby (2) (3)
(1) Department of Mathematics [Lowell]
(2) University of Connecticut
(3) University of Connecticut [Storrs]
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Imported on
November 21, 2016
Published on
December 31, 2012
Last modified on
March 31, 2025
Volume
DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Proceedings
DOI
10.46298/dmtcs.2356
License
https://about.hal.science/hal-authorisation-v1
National Science Foundation
1001905
Deterministic analogues of random processes
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744
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884
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Homomesy in products of two chains

James Propp (1), Tom Roby (2) (3)
(1) Department of Mathematics [Lowell]
(2) University of Connecticut
(3) University of Connecticut [Storrs]
Abstract
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.
Keywords
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • antichains
  • combinatorial ergodicity
  • homomesy
  • orbit
  • order ideals
  • poset
  • product of chains
  • promotion
  • rowmotion
  • sandpile
  • toggle group.
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