The expected jaggedness of order ideals

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-López-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.

20 pages, 7 figures

References in corpus (3)

• Genera of Brill-Noether curves and staircase paths in Young tableaux
• A historical survey of P-partitions
• Spectral gap for random-to-random shuffling on linear extensions

Cited by in corpus (10)

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• Rowmotion and Increasing Labeling Promotion
• On Order Ideals of Minuscule Posets III: The CDE Property
• The CDE property for minuscule lattices
• Poset edge densities, nearly reduced words, and barely set-valued tableaux
• Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion
• The CDE property for skew vexillary permutations
• The birational Lalanne-Kreweras involution
• On the $q$-Enumeration of Barely Set-Valued Tableaux and Plane Partitions
• Homomesy via Toggleability Statistics