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arXiv · 2202.07981

m-Nearly k-Universal Words -- Investigating Simon Congruence

Pamela Fleischmann·Lukas Haschke·Annika Huch·Annika Mayrock·Dirk Nowotka
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Abstract

Determining the index of the Simon congruence is a long outstanding open problem. Two words $u$ and $v$ are called Simon congruent if they have the same set of scattered factors, which are parts of the word in the correct order but not necessarily consecutive, e.g., $\mathtt{oath}$ is a scattered factor of $\mathtt{logarithm}$. Following the idea of scattered factor $k$-universality, we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered factors of length $k$ are absent, w.r.t. Simon congruence. We present a full characterisation as well as the index of the congruence for $m=1$. For $m\neq 1$, we show some results if in addition $w$ is $(k-1)$-universal as well as some further insights for different $m$.

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Pamela Fleischmann, Lukas Haschke, Annika Huch, Annika Mayrock, Dirk Nowotka. 2022-02-16. m-Nearly k-Universal Words -- Investigating Simon Congruence. https://arxiv.org/abs/2202.07981

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