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.

翻译：\emph{回文}是指正读反读均相同的单词。\emph{块回文分解}（或称\emph{BP-分解}）是将单词分解为若干块，使得每个相同块替换为不同符号后形成回文。我们将BP-分解中的块数称为BP-分解的\emph{宽度}。单词$w$的\emph{最大BP-分解}是具有最大宽度的$w$的BP-分解。我们研究具有特定BP-分解的单词。首先，我们给出长度为$n$且最大BP-分解宽度为$t$的单词数量的递推式。其次，我们证明单词最大BP-分解的期望宽度趋于常数。第三，我们给出BP-分解的另一极值变体——\emph{最小BP-分解}的一些结果。单词$w$的\emph{边界}是同时作为$w$的真前缀和真后缀的非空单词。最后，我们通过展示具有唯一边界的单词与其最小和最大BP-分解相重合的单词之间的联系得出结论。