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

Journal of Integer Sequences, Vol. 26 (2023), Article 23.7.3 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.