Asymptotics of multinomial sums and identities between multi-integrals

We calculate the asymptotics of combinatorial sums Sigma(alpha)f(alpha)((n) (alpha))(beta),where alpha = (alpha(I) ,...,alpha(h)) With alpha(i) = alpha (j) for certain i, j. Here h is fixed and the alpha(i)'s are natural number s. This implies the asymptotics of the corresponding S-n-character degrees Sigma(lambda)f(lambda)d(lambda)(beta). For certain sequences of S-n charact ers which involve Young's rule, the latter asymptotics were obtained earlie r [I] by a different method. Equating the two asymptotics, we obtain equati ons between multi-integrals which involve Gaussian measures. Special cases here give certain extensions of the Mehta integral [5], [6].