Rich Words in the Block Reversal of a Word

TL;DR

This paper characterizes binary and non-binary words whose block reversal contains only rich words, establishing conditions on the length of the run sequence and providing structural insights.

Contribution

It introduces a characterization of words with rich block reversals, especially for binary words, based on the run sequence length and structural properties.

Findings

Richness depends on run sequence length for binary words.

All elements are rich only if run sequence length is between 2 and 8.

Structural description of words with rich block reversals.

Abstract

The block reversal of a word $w$, denoted by $\mathtt{BR}(w)$, is a generalization of the concept of the reversal of a word, obtained by concatenating the blocks of the word in the reverse order. We characterize non-binary and binary words whose block reversal contains only rich words. We prove that for a binary word $w$, richness of all elements of $\mathtt{BR}(w)$ depends on $l(w)$, the length of the run sequence of $w$. We show that if all elements of $\mathtt{BR}(w)$ are rich, then $2\leq l(w)\leq 8$. We also provide the structure of such words.