$\alpha$-$\beta$-Factorization and the Binary Case of Simon's Congruence
Abstract
In 1991 H\'ebrard introduced a factorization of words that turned out to be a powerful tool for the investigation of a word's scattered factors (also known as (scattered) subwords or subsequences). Based on this, first Karandikar and Schnoebelen introduced the notion of -richness and later on Barker et al. the notion of -universality. In 2022 Fleischmann et al. presented a generalization of the arch factorization by intersecting the arch factorization of a word and its reverse. While the authors merely used this factorization for the investigation of shortest absent scattered factors, in this work we investigate this new --factorization as such. We characterize the famous Simon congruence of -universal words in terms of -universal words. Moreover, we apply these results to binary words. In this special case, we obtain a full characterization of the classes and calculate the index of the congruence. Lastly, we start investigating the ternary case, present a full list of possibilities for -factors, and characterize their congruence.
Cite
@article{arxiv.2306.14192,
title = {$\alpha$-$\beta$-Factorization and the Binary Case of Simon's Congruence},
author = {Pamela Fleischmann and Jonas Höfer and Annika Huch and Dirk Nowotka},
journal= {arXiv preprint arXiv:2306.14192},
year = {2023}
}