On the number of partition weights with Kostka multiplicity one Zachary Gates∗ Brian Goldmany Department of Mathematics Mathematics Department University of Virginia College of William and Mary Charlottesville, VA Williamsburg, VA
[email protected] [email protected] C. Ryan Vinrootz Mathematics Department College of William and Mary Williamsburg, VA
[email protected] Submitted: Jul 24, 2012; Accepted: Dec 19, 2012; Published: Dec 31, 2012 Mathematics Subject Classification: 05A19 Abstract 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 f0 < 1¯ < 1 < 2¯ < 2 < · · · g, and such that i and ¯i contribute equally to the weight. For a weight (w0; µ) with µ a partition, the signed Kostka number K± is defined as the number of semistandard signed tableaux λ,(w0,µ) ± of shape λ and weight (w0; µ), and J (λ) is then defined to be the number of weights ± (w0; µ) such that K = 1. Using different methods than in the unsigned case, λ,(w0,µ) we find that the only nonzero value which J ±(λ) can take is 1, and we find all sequences of partitions with this property.