Counting words with Laguerre series

TL;DR

This paper introduces a novel method for counting constrained words using Laguerre series, providing a combinatorial interpretation and generating functions for specific pattern-avoiding words.

Contribution

It develops a new technique linking Laguerre polynomials to combinatorial word counting and derives explicit generating functions for pattern-avoiding words.

Findings

Derived generating functions for words avoiding vincular patterns with ones.

Established a combinatorial interpretation for Laguerre polynomial products.

Provided computational methods for counting constrained words.

Abstract

We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter \alpha = -1. We describe how such a series can be computed by finding an appropriate ordinary generating function and applying a certain transformation. We use this technique to find the generating function for the number of k-ary words avoiding any vincular pattern that has only ones, as well as words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length.