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The jaggedness of an order ideal
in a poset
is the number of maximal elements in
plus the number of minimal elements of
. 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.
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