  A correlation is a binary vector that encodes all possible positions of
overlaps of two words, where an overlap for an ordered pair of words (u,v)
occurs if a suffix of word u matches a prefix of word v. As multiple pairs can
have the same correlation, it is relevant to count how many pairs of words
share the same correlation depending on the alphabet size and word length n. We
exhibit recurrences to compute the number of such pairs -- which is termed
population size -- for any correlation; for this, we exploit a relationship
between overlaps of two words and self-overlap of one word. This theorem allows
us to compute the number of pairs with a longest overlap of a given length and
to show that the expected length of the longest border of two words
asymptotically diverges, which solves two open questions raised by Gabric in
2022. Finally, we also provide bounds for the asymptotic of the population
ratio of any correlation. Given the importance of word overlaps in areas like
word combinatorics, bioinformatics, and digital communication, our results may
ease analyses of algorithms for string processing, code design, or genome
assembly.
