方言和变式模式统计 (Pattern statistics in faro words and permutations)

We study the distribution and the popularity of some patterns in {\em $k$-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.

翻译：我们用 $1, 2,\ldots, k ⁇ c$ 来研究某些模式的分布和受欢迎程度, 即字母上的单词 $1, 2, 2,\ldots, k ⁇ c$, 通过对两个非减序长度的单词的字母进行互换而获得的。我们对这些单词和分散的 Dyck 路径( 即, Motzkin 路径, 包含$x$- 轴上的所有级步骤) 和一定数量的峰值进行双截图, 我们展示了远方文字连续模式的统计数据如何被其他路径模式统计的线性组合。然后, 我们通过提供多变量生成函数来生成长度模式在最多三个位置的分布和普及性来推断出数字结果。最后, 我们考虑一些有趣的小词汇的子类, 它们是变异、变、变、变、变变、变、变、变、变或子单词。