A natural bijection for contiguous pattern avoidance in words

TL;DR

This paper establishes a natural bijection between words avoiding certain patterns based on border lengths, providing a combinatorial proof that complements previous generating function approaches.

Contribution

It introduces an explicit bijective proof for pattern avoidance equivalence based on border lengths, advancing combinatorial understanding.

Findings

Bijection exists for words avoiding patterns with the same set of proper borders.

The bijection is explicit and combinatorial, not relying on generating functions.

The result applies to all pairs of patterns with identical border length sets.

Abstract

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 $q$.