  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.
