Polynomial Expressions for the Dimensions of the Representations of Symmetric Groups and Restricted Standard Young Tableaux

2024/10/22 by Cohen, Avichai, Shaul Zemel, Zemel, Shaul · 1 citation
Mathematics · #05A17 #05A19 #05E10 #20C30 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2410.16758

openalex publication_date 2024/10/22 · openalex created_date 2024/11/13 · openalex updated_date 2026/07/28

Abstract

Given a partition λ of a number k, it is known that by adding a long line of length n-k, the dimension of the associated representation of S_n is an integer-valued polynomial of degree k in n. We show that its expansion in the binomial basis is bounded by the length of λ, and that the resulting coefficient of index h, with alternating signs, counts the standard Young tableaux of shape λ in which a given collection of consecutive h numbers lie in increasing rows. We also construct bijections in order to demonstare explicitly that this number is indeed independent of the set of consecutive h numbers used.

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