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 [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], 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.

Keywords

• 05E18
• 06A07 (primary)
• 14Q05 (secondary)