On longest increasing subsequences in words in which all multiplicities are equal

Abstract

Gessel's famous Bessel determinant formula gives the generating function of the number of permutations without increasing subsequences of a given length. Ekhad and Zeilberger proposed the challenge of finding a suitable generalization for permutations of multisets in which all multiplicities are equal, that is, to count words of length $rn$ from an alphabet consisting of $n$ letters in which each letter appears exactly $r$ times and which have no increasing subsequences of length $d$. In this paper we present such a generating function expressible as a multiple integral of the product of a Gessel-type Toeplitz determinant with the exponentiated cycle index polynomial of the symmetric group on $r$ elements.