Borders, Palindrome Prefixes, and Square Prefixes

Daniel Gabric, Jeffrey Shallit

We show that the number of length-$n$ words over a $k$-letter alphabet having no even palindromic prefix is the same as the number of length-$n$ unbordered words, by constructing an explicit bijection between the two sets. A similar result holds for those words having no odd palindromic prefix, again by constructing a certain bijection. Using known results on borders, it follows that the number of length-$n$ words having no even (resp., odd) palindromic prefix is asymptotically $\gamma_k \cdot k^n$ for some positive constant $\gamma_k$. We obtain an analogous result for words having no nontrivial palindromic prefix. Finally, we obtain similar results for words having no square prefix, thus proving a 2013 conjecture of Chaffin, Linderman, Sloane, and Wilks.

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