最小与最大块回文分解
Smallest and Largest Block Palindrome Factorizations
AI总结：
本文研究词的块回文分解的极值问题，给出最大宽度为t的长度n词数的递推关系，证明最大BP-分解期望宽度趋于常数，并揭示唯一边界词与最小最大BP-分解重合之间的联系。
摘要：
回文是指正读和反读都相同的词。块回文分解（BP-分解）是将一个词分解为若干块，使得当每个相同的块被替换为不同的符号时，该分解成为回文。我们称BP-分解中块的数量为该BP-分解的宽度。词w的最大BP-分解是w的宽度最大的BP-分解。我们研究具有特定BP-分解的词。首先，我们给出了长度为n且最大BP-分解宽度为t的词的数量所满足的递推关系。其次，我们证明一个词的最大BP-分解的期望宽度趋于一个常数。第三，我们给出了关于BP-分解的另一种极值变体——最小BP-分解的一些结果。词w的边界是一个非空词，它既是w的真前缀又是w的真后缀。最后，我们通过展示具有唯一边界的词与最小和最大BP-分解重合的词之间的联系来结束全文。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
