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arXiv · 1507.00249

The expected jaggedness of order ideals

Melody Chan·Shahrzad Haddadan·Sam Hopkins·Luca Moci
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Abstract

The jaggedness of an order ideal I in a poset P is the number of maximal elements in I plus the number of minimal elements of P not in I. A probability distribution on the set of order ideals of P is toggle-symmetric if for every p in P, the probability that p is maximal in I equals the probability that p is minimal not in I. In this paper, we prove a formula for the expected jaggedness of an order ideal of P under any toggle-symmetric probability distribution when P is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan-L\'opez-Pflueger-Teixidor, who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp-Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

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Melody Chan, Shahrzad Haddadan, Sam Hopkins, Luca Moci. 2015-07-01. The expected jaggedness of order ideals. https://doi.org/10.1017/fms.2017.5

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