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# Homomesy in products of two chains

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.
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Conference papers

Cited literature [10 references]

https://hal.inria.fr/hal-01229696
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Submitted on : Tuesday, November 17, 2015 - 10:20:06 AM
Last modification on : Tuesday, August 6, 2019 - 11:38:37 AM
Long-term archiving on: : Thursday, February 18, 2016 - 11:40:46 AM

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dmAS0180.pdf
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### Citation

James Propp, Tom Roby. Homomesy in products of two chains. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.945-956, ⟨10.46298/dmtcs.2356⟩. ⟨hal-01229696⟩

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