Swapping Compatible Words

Source abstract

In 1981 Guibas and Odlyzko showed that the set $A$ of all words of a given length $n$ that avoid a given word $p$, and the set $B$ of all words of length $n$ that avoid a given word $q$, have the same cardinality iff $p$ and $q$ have the same self-border lengths. In 2024 Carrigan et. al. asked if there is a "natural" bijection between $A$ and $B$. This paper makes partial progress toward answering that question. We provide such a bijection by imposing additional conditions on $p$ and $q$, but we also obtain a stronger conclusion: there is a "natural" bijection $Φ$ from the set of all finite words to itself such that a word $Ω$ has a $p$ or $q$ at a given position iff $Φ(Ω)$ has, repectively, a $q$ or a $p$ at that same position. An algorithm is given for converting $Ω$ into $Φ(Ω)$.