Proof of a Conjecture of Reiner--Tenner--Yong on Barely Set-Valued Tableaux

Abstract

The notion of a barely set-valued semistandard Young tableau was introduced by Reiner, Tenner and Yong in their study of the probability distribution of edges in the Young lattice of partitions. Given a partition λ and a positive integer k, let BSSYT(λ, k) (respectively, SYT(λ, k)) denote the set of barely setvalued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape λ with entries in row i not exceeding k + i. In the case when λ is a rectangular staircase partition δ d (b a ), Reiner, Tenner and Yong conjectured that |BSSYT(λ, k)| = kab(d−1) (a+b) |SYT(λ, k)|. In this paper, we establish a connection between barely set-valued tableaux and reverse plane partitions with designated corners. We show that for any shape λ, the expected jaggedness of a subshape of λ under the weak probability distribution can be expressed as 2|BSSYT(λ,k)| k|SYT(λ,k)| . On the other hand, when λ is a balanced shape with r rows and c columns, Chan, Haddadan, Hopkins and Moci proved that the expected jaggedness of a subshape in λ under the weak distribution equals 2rc/(r + c). Hence, for a balanced shape λ with r rows and c columns, we establish the relation that |BSSYT(λ, k)| = krc (r+c) |SYT(λ, k)|. Since a rectangular staircase shape δ d (b a ) is a balanced shape, we confirm the conjecture of Reiner, Tenner and Yong.

On the $q$-Enumeration of Barely Set-Valued Tableaux and Plane Partitions

“…Independently of this work in Brill-Noether theory, Reiner, Tenner, and Yong [25] also investigated barely set-valued tableaux from the more traditional vantage point of symmetric functions. They proved a number of results extending (1.4), and made a number of conjectures, and subsequently there has been a reasonable amount of research devoted to counting classes of barely set-valued tableaux [18,12,19]. We note that while (1.1) applies to any partition, product formulas like (1.4) for #SYT +1 (λ) exist only for very special shapes λ like rectangles.…”