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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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https://hal.inria.fr/hal-01229696
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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. ⟨hal-01229696⟩

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