Pattern statistics in faro words and permutations
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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.
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