Forbidden substrings on weighted alphabets

TL;DR

This paper extends the classical combinatorial problem of counting strings avoiding certain substrings to weighted alphabets, inspired by integer composition avoidance and random walk considerations.

Contribution

It generalizes Guibas and Odlyzko's generating function approach to weighted alphabets, broadening the scope of substring avoidance enumeration.

Findings

Derived a generating function for weighted alphabet strings avoiding forbidden substrings.

Connected substring avoidance to integer composition problems and random walks.

Extended classical combinatorial enumeration methods to weighted cases.

Abstract

In an influential 1981 paper, Guibas and Odlyzko constructed a generating function for the number of length n strings over a finite alphabet that avoid all members of a given set of forbidden substrings. Here we extend this result to the case in which the strings are weighted. This investigation was inspired by the problem of counting compositions of an integer n that avoid all compositions of a smaller integer m, a notion which arose from the consideration of one-sided random walks.