Rooks on ferrers boards and matrix integrals

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.