Rooks on Ferrers boards and matrix integrals

S. V. Kerov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 136-146

Résumé

Let $C(n,N)=\int_{H_N}\operatorname{tr}Z^{2n}\,\mu(dZ)$ denote a matrix integral by a $U(N)$-invariant gaussian measure $\mu$ on the space $H_N$ of hermitian $N\times{N}$ 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 $$ C(n,N) = (2n - 1)!! \sum_{k=0}^n \binom N{k+1}\binom nk\, 2^k $$ found by J. Harer and D. Zagier follows from our interpretation immediately.