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

TL;DR

This paper investigates the properties of words with specific scattered factor constraints under Simon congruence, providing characterizations and index calculations for certain cases, advancing understanding of the open problem.

Contribution

It offers a full characterization and index determination for 1-nearly k-universal words under Simon congruence, and explores properties for other values of m.

Findings

Characterization of 1-nearly k-universal words

Index of Simon congruence for m=1

Results for m ≠ 1 with additional assumptions

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 $m$.