Two words $p$ and $q$ are avoided by the same number of length-$n$ words, for all $n$, precisely when $p$ and $q$ have the same set of border lengths. Previous proofs of this theorem use generating functions but do not provide an explicit bijection. We give a bijective proof for all pairs $p, q$ that have the same set of proper borders, establishing a natural bijection from the set of words avoiding $p$ to the set of words avoiding $q$.

翻译：对于所有长度$n$的单词，当且仅当两个单词$p$和$q$具有相同的边界长度集合时，它们被相同数量的长度为$n$的单词所避免。先前对该定理的证明使用了生成函数，但未提供显式双射。我们为所有具有相同正确边界集合的单词对$p, q$给出了一个双射证明，建立了一个从避免$p$的单词集合到避免$q$的单词集合的自然双射。