The locality of words is a relatively young structural complexity measure, introduced by Day et al. in 2017 in order to define classes of patterns with variables which can be matched in polynomial time. The main tool used to compute the locality of a word is called marking sequence: an ordering of the distinct letters occurring in the respective order. Once a marking sequence is defined, the letters of the word are marked in steps: in the ith marking step, all occurrences of the ith letter of the marking sequence are marked. As such, after each marking step, the word can be seen as a sequence of blocks of marked letters separated by blocks of non-marked letters. By keeping track of the evolution of the marked blocks of the word through the marking defined by a marking sequence, one defines the blocksequence of the respective marking sequence. We first show that the words sharing the same blocksequence are only loosely connected, so we consider the stronger notion of extended blocksequence, which stores additional information on the form of each single marked block. In this context, we present a series of combinatorial results for words sharing the extended blocksequence.

中文翻译：

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k-local Words的块序列

词的局部性是一种相对年轻的结构复杂性度量，由 Day 等人引入。在 2017 年，为了定义可以在多项式时间内匹配的变量的模式类。用于计算单词位置的主要工具称为标记序列：按相应顺序出现的不同字母的排序。一旦定义了标记序列，单词的字母就会被分步标记：在第 i 个标记步骤中，标记序列的第 i 个字母的所有出现都被标记。因此，在每个标记步骤之后，单词可以被视为由未标记字母块分隔的标记字母块序列。通过通过标记序列定义的标记跟踪单词的标记块的演变，可以定义相应标记序列的块序列。我们首先表明共享相同块序列的单词只是松散连接，因此我们考虑扩展块序列的更强概念，它存储有关每个单个标记块形式的附加信息。在这种情况下，我们为共享扩展块序列的单词提供了一系列组合结果。