Swapping Compatible Words
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).
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BibTeX
@misc{naimi2026swapping,
title = {{Swapping Compatible Words}},
author = {Ramin Naimi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31967},
eprint = {2610.31967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31967},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Swapping Compatible Words
AU - Ramin Naimi
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31967
UR - https://arxiv.org/abs/2610.31967
AB - 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).
ER -