Avoiding substrings in compositons

TL;DR

This paper extends classical substring avoidance results to compositions, providing generating functions for avoiding certain substrings while considering composition length and weight, and offering explicit formulas for specific cases.

Contribution

It introduces a novel analogue of substring avoidance results for compositions, incorporating length and weight constraints, and generalizes recent findings by Myers.

Findings

Derived generating functions for compositions avoiding specific substrings.

Provided explicit formulas for certain families of avoided substrings.

Extended classical substring avoidance results to composition enumeration with constraints.

Abstract

A classical result by Guibas and Odlyzko obtained in 1981 gives the generating function for the number of strings that avoid a given set of substrings with the property that no substring is contained in any of the others. In this paper, we give an analogue of this result for the enumeration of compositions that avoid a given set of prohibited substrings, subject to the compositions' length (number of parts) and weight. We also give examples of families of strings to be avoided that allow for an explicit formula for the generating function. Our results extend recent results by Myers on avoidance of strings in compositions subject to weight, but not length.