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 ∈ 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.

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

“…To proceed further we need to introduce toggling and the "toggle perspective." This perspective was first developed in [12], while the "toggle" terminology comes from Striker and Williams [88]. For p ∈ P we define the toggle at p τ p : J(P ) → J(P ) as The τ p are involutions.…”

“…Finding such a linear equation is essentially the only the general tool we have for proving that a distributive lattice is CDE. Theorem 3.7 was first proved in a case-by-case manner in [35], also using results from [12]. Shortly thereafter, Rush [69] gave a uniform proof of Theorem 3.7.…”

“…• P is the poset corresponding to a "balanced skew shape" (see [12]); • P is the poset corresponding to a "shifted-balanced shape" (see [35]).…”

“…Namely, we have: Proof. The cases (Λ OG (6,12) , Φ + (H 3 )) and (Λ Q 2n , Φ + (I 2 (2n))) are simple exercises. For (P, Q) = (Λ Gr(k,n) , T k,n ), we want a bijection between J(P ) and J(Q) which preserves down-degree.…”

“…Conjecture 3.19. Let (P, Q) ∈ {(Λ Gr(k,n) , T k,n ), (Λ OG (6,12) , Φ + (H 3 )), (Λ Q 2n , Φ + (I 2 (2n)))} be a minuscule doppelgänger pair. Then #PP ℓ e (P ) = #PP ℓ e (Q) for any ℓ ≥ 1, e ≥ 0.…”