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 words obtained by interlacing the letters of the two nondecreasing $k$-ary words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths with a given number of peaks. We show how the bijection maps statistics of consecutive patterns into linear combinations of other pattern statistics on paths. 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：我们研究某些模式的分布和受欢迎程度,这些模式通过将两个未降息但以美元计的长度的字句相互交错而获得。我们用给定的峰值表示这些字词和分散的Dyck路径之间的两边。我们展示了连续模式的两边图案统计数据如何成为路径上其他模式统计的线性组合。我们通过提供多变量生成函数来推断出数字结果, 并在最多三个长度模式的分布和受欢迎程度。最后, 我们考虑了一些有趣的次类, 包括变异、变异、变异、变异、变异或次异的词。