Borders, palindrome prefixes, and square prefixes

We show that the number of length- n words over a k -letter alphabet having no even palindromic prefix is same as unbordered words, by constructing an explicit bijection between two sets. A slightly different but analogous result holds for those odd prefix. Using known results on borders, we get asymptotic enumeration (resp., odd) obtain nontrivial Finally, similar square prefix, thus proving 2013 conjecture Chaffin, Linderman, Sloane, and Wilks. • Finds new connection borders palindrome prefixes. Uses this to with various sizes Solves open problem