# Smallest and Largest Block Palindrome Factorizations

## TL;DR

This paper investigates the properties of block palindrome factorizations in words, providing recurrence relations, expected value analysis, and extremal case studies, revealing structural insights and connections to borders in words.

## Contribution

It introduces new recurrence formulas, analyzes the expected width of largest BP-factorizations, and explores the relationship between smallest and largest BP-factorizations and word borders.

## Key findings

- Recurrence for the number of words with a given largest BP-factorization width.
- Expected width of largest BP-factorization tends to a constant.
- Connection between words with a unique border and coinciding smallest and largest BP-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 $w$ is the BP-factorization of $w$ with the maximum width. We study words with certain BP-factorizations. First, we give a recurrence for the number of length-$n$ words with largest BP-factorization of width $t$. 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 $w$ is a non-empty word that is both a proper prefix and suffix of $w$. Finally, we conclude by showing a connection between words with a unique border and words whose smallest and largest BP-factorizations coincide.