Jumbled Scattered Factors

TL;DR

This paper introduces the concept of jumbled scattered factors, combining ideas from scattered factors and jumbled words, and explores their properties, characterizations, and relations to classical string problems.

Contribution

It defines and characterizes jumbled scattered factors, relating them to longest common subsequence and Simon's congruence, advancing understanding of word factorization with jumbling.

Findings

Characterization of jumbled scattered factors based on the number of jumbled letters

Relation between longest common subsequence and jumbled factors

Analysis of minimal jumbles needed for factorization

Abstract

In this work, we combine the research on (absent) scattered factors with the one of jumbled words. For instance, $\mathtt{wolf}$ is an absent scattered factor of $\mathtt{cauliflower}$ but since $\mathtt{lfow}$, a jumbled (or abelian) version of $\mathtt{wolf}$, is a scattered factor, $\mathtt{wolf}$ occurs as a jumbled scattered factor in $\mathtt{cauliflower}$. A \emph{jumbled scattered factor} $u$ of a word $w$ is constructed by letters of $w$ with the only rule that the number of occurrences per letter in $u$ is smaller than or equal to the one in $w$. We proceed to partition and characterise the set of jumbled scattered factors by the number of jumbled letters and use the latter as a measure. For this new class of words, we relate the folklore longest common subsequence (scattered factor) to the number of required jumbles. Further, we investigate the smallest possible number of jumbles alongside the jumbled scattered factor relation as well as Simon's congruence from the point of view of jumbled scattered factors and jumbled universality.