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 slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words having no even (resp., odd) palindromic prefix. 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.

通过显示两组之间的显式双射，我们证明了在没有偶数回文前缀的k字母上，长度为n的单词的数量与长度为n的无边界单词的数量相同。对于那些没有奇数回文前缀的单词，存在略有不同但类似的结果。使用边界上的已知结果，我们得到了无偶数（resp。，奇数）回文前缀的单词数量的渐近枚举。对于没有非平凡回文前缀的单词，我们获得了类似的结果。最后，对于没有平方前缀的单词，我们获得了相似的结果，从而证明了2013年对查芬，林德曼，斯隆和威尔克斯的猜想。