The expected jaggedness of order ideals

The jaggedness of an order ideal in a poset is the number of maximal elements in plus the number of minimal elements of not in . A probability distribution on the set of order ideals of is toggle-symmetric if for every , the probability that is maximal in equals the probability that is minimal not in . In this paper, we prove a formula for the expected jaggedness of an order ideal of under any toggle-symmetric probability distribution when 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.