THE EXPECTED JAGGEDNESS OF ORDER IDEALS

Melody Chan,Shahrzad Haddadan,Luca Moci,Sam Hopkins

doi:10.1017/fms.2017.5

Abstract

Thejaggednessof 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$P$istoggle-symmetricif 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.

Highlights

The six lattice paths in a 2 × 2 grid occur in μlin with the probabilities shown below: We define the jaggedness of such a lattice path to be its number of turns
We show how certain conditions on Tp imply conditions on jag, as in the following main definition of the section
Let be a weak reverse P-partition of height m; we extend to P by setting (0) := 0 and (1) := m

Summary

Introduction

The six lattice paths in a 2 × 2 grid occur in μlin with the probabilities shown below: We define the jaggedness of such a lattice path to be its number of turns This is the same as the jaggedness of the order ideal to its northwest (see Definition 2.1). The expected jaggedness of a lattice path in an a ×b grid, chosen under the distribution μlin, is exactly 2ab/(a + b), the harmonic mean of a and b. Our result makes explicit that the only dependence is on the outer corners and their displacements This will allow us to immediately derive that for any balanced shape, the expected jaggedness is always the harmonic mean; see Corollary 3.8.

Toggle-symmetric distributions

The expected jaggedness in skew shapes

Open questions