论文标题
Faro单词和排列中的模式统计数据
Pattern statistics in faro words and permutations
论文作者
论文摘要
我们研究了某些模式在$ k $ - ary faro单词中的分布和流行，即字母$ \ \ {1，2，\ ldots，k \} $，通过交错了两个不差的长度差异的字母，从而获得了最多一个。我们介绍了这些单词和分散的Dyck路径（即具有给定数量峰的$ x $ -Axis上所有级别步骤的Motzkin路径）。我们展示了如何将Faro单词连续模式的统计数据映射到路径上其他模式统计的线性组合中。然后，我们通过为最多三个长度模式的分布和流行提供多元生成功能来推断枚举结果。最后，我们考虑了一些有趣的Faro单词的子类，这些单词是排列，交流，毁灭或次要单词。
We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the $x$-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Finally, we consider some interesting subclasses of faro words that are permutations, involutions, derangements, or subexcedent words.
