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- W4298031806 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 $lambda$ and a positive integer $k$, let ${mathrm{BSSYT}}(lambda,k)$ (respectively, ${mathrm{SYT}}(lambda,k)$) denote the set of barely set-valued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape $lambda$ with entries in row $i$ not exceeding $k+i$. In the case when $lambda$ is a rectangular staircase partition $delta_d(b^a)$, Reiner, Tenner and Yong conjectured that $|{mathrm{BSSYT}}(lambda,k)|= frac{kab(d-1)}{(a+b)} |{mathrm{SYT}}(lambda,k)|$. 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 $lambda$, the expected jaggedness of a subshape of $lambda$ under the weak probability distribution can be expressed as $frac{2|{mathrm{BSSYT}}(lambda,k)|} {k|{mathrm{SYT}}(lambda,k)|}$. On the other hand, when $lambda$ is a balanced shape with $r$ rows and $c$ columns, Chan, Haddadan, Hopkins and Moci proved that the expected jaggedness of a subshape in $lambda$ under the weak distribution equals $2rc/(r+c)$. Hence, for a balanced shape $lambda$ with $r$ rows and $c$ columns, we establish the relation that $|{mathrm{BSSYT}}(lambda,k)|=frac{krc}{(r+c)}|{mathrm{SYT}}(lambda,k)|$. Since a rectangular staircase shape $delta_d(b^a)$ is a balanced shape, we confirm the conjecture of Reiner, Tenner and Yong." @default.
- W4298031806 created "2022-10-01" @default.
- W4298031806 creator A5035925564 @default.
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- W4298031806 date "2018-07-22" @default.
- W4298031806 modified "2023-09-26" @default.
- W4298031806 title "Proof of a Conjecture of Reiner-Tenner-Yong on Barely Set-valued Tableaux" @default.
- W4298031806 doi "https://doi.org/10.48550/arxiv.1807.08292" @default.
- W4298031806 hasPublicationYear "2018" @default.
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