Scattered Factor Universality -- The Power of the Remainder
This work addresses incremental theoretical problems in formal language theory, specifically for researchers in combinatorics on words.
The paper tackles two open problems in scattered factor universality, generalizing a main theorem to arbitrary alphabets and modifying another to characterize circular universality through universality, while providing insights into the remainder of arch factorisation under word repetitions.
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.