Counting and Verifying Abelian Border Arrays of Binary Words

In this note, we consider the problem of counting and verifying abelian border arrays of binary words. We show that the number of valid abelian border arrays of length $n$ is $2^{n-1}$. We also show that verifying whether a given array is the abelian border array of some binary word reduces to computing the abelian border array of a specific binary word. Thus, assuming the word-RAM model, we present an $O\left(\frac{n^2}{\log^2n}\right)$ time algorithm for the abelian border array verification problem.