COMBINATORIAL CONSTRUCTIONS FOR INTEGRALS OVER NORMALLY DISTRIBUTED RANDOM MATRICES

Recent results of Hanlon, Stanley, and Stembridge give the expected va lues of certain functions of matrices of normal variables in the real and complex cases. They point out that in both cases the results are e quivalent to combinatorial results and suggest further that these resu lts may have purely combinatorial proofs, in this way avoiding the use of the theory of spherical functions. Such proofs are given in this p aper. In the complex case we use the familiar cycle decomposition for permutations. In the real case we introduce an analogous decomposition into cyclically ordered sequences, called chains, which makes the rea l and complex cases strikingly similar.