Locally restricted compositions over a finite group

Abstract

One may generalize integer compositions by replacing the positive integers with a different additive semigroup, giving the broader concept of a “composition over a semigroup”. Here we focus on semigroups which are finite groups and achieve asymptotic enumeration of compositions over a finite group which satisfy a local restriction. These compositions are associated to walks on a voltage graph whose structure is exploited to simplify asymptotic expressions. Specifically, we show that under mild conditions the number of locally restricted compositions of a group element is asymptotically independent of the particular group element. We apply this result to subword pattern avoidance and other examples such as generalized Carlitz compositions.