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

TL;DR

This paper generalizes Gessel's determinant formula to count words with equal letter multiplicities that lack long increasing subsequences, providing a new integral representation involving Toeplitz determinants and symmetric group polynomials.

Contribution

It introduces a novel generating function for words with equal letter multiplicities avoiding long increasing subsequences, extending Gessel's classical permutation result.

Findings

Provides a multiple integral expression for the generating function

Extends Gessel's formula to multisets with equal multiplicities

Connects combinatorial enumeration with Toeplitz determinants and symmetric group polynomials

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.