Scattered Factor Universality -- The Power of the Remainder

TL;DR

This paper extends the theory of scattered factor universality to arbitrary alphabets and characterizes circular universality through universality, providing new insights into the behavior of the arch factorization's remainder.

Contribution

It generalizes key theorems of scattered factor universality to broader contexts and offers a new characterization linking circular and standard universality.

Findings

Generalized universality theorems to arbitrary alphabets

Characterized circular universality via universality

Analyzed the behavior of the arch factorization's remainder with repetitions

Abstract

Scattered factor (circular) universality was firstly introduced by Barker et al. in 2020. A word $w$ is called $k$-universal for some natural number $k$, if every word of length $k$ of $w$'s alphabet occurs as a scattered factor in $w$; it is called circular $k$-universal if a conjugate of $w$ is $k$-universal. Here, a word $u=u_1\cdots u_n$ is called a scattered factor of $w$ if $u$ is obtained from $w$ by deleting parts of $w$, i.e. there exists (possibly empty) words $v_1,\dots,v_{n+1}$ with $w=v_1u_1v_2\cdots v_nu_nv_{n+1}$. In this work, we prove two problems, left open in the aforementioned paper, namely a generalisation of one of their main theorems to arbitrary alphabets and a slight modification of another theorem such that we characterise the circular universality by the universality. On the way, we present deep insights into the behaviour of the remainder of the so called arch factorisation by Hebrard when repetitions of words are considered.