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Rooks on ferrers boards and matrix integrals

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Abstract

Let\(C(n, N) = \smallint _{H_N } tr Z^{2n} \mu (dZ)\) denote a matrix integral over a U(N)- invariant Gaussian measure Μ on the space Hn of Hermitian N×N matrices. The integral is known to be always a positive integer. We derive a simple combinatorial interpretation of this integral in terms of rook configurations on Ferrers boards. The formula

$$C(n,N) = (2n - 1)!! \sum\limits_{k = 0}^n {\left( {\mathop {k + 1}\limits^N } \right) \left( {\mathop k\limits^n } \right) } 2^k ,$$

found by J. Harer and D. Zagier, immediately follows from our interpretation.

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Authors
  1. S. V. Kerov
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Additional information

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 136–146.

Partially supported by grants RFFI-96-01-00676 and INTAS-94-3420.

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Kerov, S.V. Rooks on ferrers boards and matrix integrals. J Math Sci 96, 3531–3536 (1999). https://doi.org/10.1007/BF02175831

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  • DOI: https://doi.org/10.1007/BF02175831

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