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We show that the number of length-n words over a k-letter alphabet having noeven palindromic prefix is the same as the number of length-n unbordered words,by constructing an explicit bijection between the two sets. A slightlydifferent but analogous result holds for those words having no odd palindromicprefix. Using known results on borders, we get an asymptotic enumeration forthe number of words having no even (resp., odd) palindromic prefix . We obtainan analogous result for words having no nontrivial palindromic prefix. Finally,we obtain similar results for words having no square prefix, thus proving a2013 conjecture of Chaffin, Linderman, Sloane, and Wilks.

Borders, Palindrome Prefixes, and Square Prefixes