Smallest and Largest Block Palindrome Factorizations
Abstract
A \emph{palindrome} is a word that reads the same forwards and backwards. A \emph{block palindrome factorization} (or \emph{BP-factorization}) is a factorization of a word into blocks that becomes palindrome if each identical block is replaced by a distinct symbol. We call the number of blocks in a BP-factorization the \emph{width} of the BP-factorization. The \emph{largest BP-factorization} of a word is the BP-factorization of with the maximum width. We study words with certain BP-factorizations. First, we give a recurrence for the number of length- words with largest BP-factorization of width . Second, we show that the expected width of the largest BP-factorization of a word tends to a constant. Third, we give some results on another extremal variation of BP-factorization, the \emph{smallest BP-factorization}. A \emph{border} of a word is a non-empty word that is both a proper prefix and suffix of . Finally, we conclude by showing a connection between words with a unique border and words whose smallest and largest BP-factorizations coincide.
Cite
@article{arxiv.2302.13147,
title = {Smallest and Largest Block Palindrome Factorizations},
author = {Daniel Gabric and Jeffrey Shallit},
journal= {arXiv preprint arXiv:2302.13147},
year = {2023}
}