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A \emph{palindrome} is a word that reads the same forwards and backwards. A\emph{block palindrome factorization} (or \emph{BP-factorization}) is afactorization of a word into blocks that becomes palindrome if each identicalblock is replaced by a distinct symbol. We call the number of blocks in aBP-factorization the \emph{width} of the BP-factorization. The \emph{largestBP-factorization} of a word $w$ is the BP-factorization of $w$ with the maximumwidth. We study words with certain BP-factorizations. First, we give arecurrence for the number of length-$n$ words with largest BP-factorization ofwidth $t$. Second, we show that the expected width of the largestBP-factorization of a word tends to a constant. Third, we give some results onanother extremal variation of BP-factorization, the \emph{smallestBP-factorization}. A \emph{border} of a word $w$ is a non-empty word that isboth a proper prefix and suffix of $w$. Finally, we conclude by showing aconnection between words with a unique border and words whose smallest andlargest BP-factorizations coincide.

Smallest and Largest Block Palindrome Factorizations