A determinant-like formula for the Kostka numbers
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formulayoungdeterminanttableauxcountingfrobeniusgeneralizationgiven
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Young tableaux are ubiquitous in various branches of mathematics. There are two counting formulas for standard Young tableaux. The first involves a determinant and goes back to Frobenius and Young, and the second is the hook formula by Frame, Robinson and Thrall. We present a generalization of the determinant formula for semistandard Young tableaux of given shape and of given content. Our counting formula -- though not a determinant -- is a generalization of the determinant formula by Frobenius and Young.
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Derivative relations for determinants, Pfaffians and characteristic polynomials in random matrix theory
Explicit expressions are proven for higher-order and mixed derivatives of determinant and Pfaffian ratios over Vandermonde determinants in random matrix theory.
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