The expected jaggedness of order ideals

TL;DR

This paper derives a formula for the expected jaggedness of order ideals in a skew Young diagram poset under toggle-symmetric distributions, extending previous combinatorial and algebraic results.

Contribution

It provides a new explicit formula for expected jaggedness in skew Young diagram posets under toggle-symmetric measures, linking combinatorics and algebraic geometry.

Findings

Derived a formula for expected jaggedness in skew Young diagrams

Extended combinatorial theorems to algebraic geometry applications

Applied results to homomesy phenomena in poset statistics

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.