Probabilistic methods are used to asymptotically estimate certain combinatorial sums. When applied to Young-derived sequences of characters of symmetric groups, these asymptotics yield interesting integration formulas.
1998 Academic Press
Probabilistic arguments are quite useful in proving asymptotic estimates for combinatorial sums. Such methods encompass the spirit of the DeMoivre Laplace central limit theorem. An example is described in Section 1 that arose in the analysis of character sequences on the symmetric group ([3]). Similar estimates have been used in the algebraic computation of Gaussian integrals of Selberg-type ([1], [3]).
The principal issue considered here is the following question: increase the lengths of the columns of Young diagrams by a fixed factor q and then determine the changes that occur in the asymptotics of the associated degrees of characters of symmetric groups. Earlier results ([1], ) and the calculation of Section 1 allow the determination of such estimates for Young-derived sequences of characters and yield in turn some rather interesting integration formulas. Some of the integrals involved are extensions of Selberg and Dyson Macdonald Mehta type integrals ([4]).