Dyck Words, Lattice Paths, and Abelian Borders

TL;DR

This paper derives exact formulas for counting binary words with specific minimal abelian border lengths using Dyck words and lattice paths, extending previous bounds and results to multiple borders and partial words.

Contribution

It introduces new formulas for counting binary words with given abelian border properties, extending prior bounds and generalizing to multiple borders and partial words.

Findings

Derived exact counts for binary words with minimal abelian borders.

Extended results to words with multiple abelian borders.

Generalized formulas to partial words.

Abstract

We use results on Dyck words and lattice paths to derive a formula for the exact number of binary words of a given length with a given minimal abelian border length, tightening a bound on that number from Christodoulakis et al. (Discrete Applied Mathematics, 2014). We also extend to any number of distinct abelian borders a result of Rampersad et al. (Developments in Language Theory, 2013) on the exact number of binary words of a given length with no abelian borders. Furthermore, we generalize these results to partial words.