2018/07/22 by Fan, Neil J. Y., Guo, Peter L., Sun, Sophie C. C.
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.08292
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 set-valued 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 δd(ba), 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 \frac2|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 δd(ba) is a balanced shape, we confirm the conjecture of Reiner, Tenner and Yong.