RUS  ENG
Full version
JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 1997 Volume 240, Pages 136–146 (Mi znsl471)

This article is cited in 8 papers

Rooks on Ferrers boards and matrix integrals

S. V. Kerov

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: 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.

UDC: 519.217+517.986

Received: 30.10.1996


 English version:
Journal of Mathematical Sciences (New York), 1999, 96:5, 3531–3536

Bibliographic databases:


© Steklov Math. Inst. of RAS, 2025