On the Number of Partition Weights with Kostka Multiplicity One

Given a positive integer n, and partitions λ and μ of n, let Kλμ denote the Kostka number, which is the number of semistandard Young tableaux of shape λ and weight μ. Let J(λ) denote the number of μ such that Kλμ = 1. By applying a result of Berenshtein and Zelevinskii, we obtain a formula for J(λ) in terms of restricted partition functions, which is recursive in the number of distinct part sizes of λ. We use this to classify all partitions λ such that J(λ) = 1 and all λ such that J(λ) = 2. We then consider signed tableaux, where a semistandard signed tableau of shape λ has entries from the ordered set {0 < 1̄ < 1 < 2̄ < 2 < · · · }, and such that i and ī contribute equally to the weight. For a weight (w0, μ) with μ a partition, the signed Kostka number K± λ,(w0,μ) is defined as the number of semistandard signed tableaux of shape λ and weight (w0, μ), and J ±(λ) is then defined to be the number of weights (w0, μ) such that K ± λ,(w0,μ) = 1. Using different methods than in the unsigned case, we find that the only nonzero value which J±(λ) can take is 1, and we find all sequences of partitions with this property. We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups. ∗Supported by NSF grant DMS-0854849. †Supported by the College of William and Mary. ‡Supported by NSF grant DMS-0854849. the electronic journal of combinatorics 19(4) (2012), #P52 1