Introduction to partially ordered patterns

Kitaev, Sergey (2007) Introduction to partially ordered patterns. Discrete Applied Mathematics, 155 (8). pp. 929-944. ISSN 0166-218X (https://doi.org/10.1016/j.dam.2006.09.011)

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Abstract

We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several new results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayana, and Pell numbers, as well as generalizations of a few results in the literature including the descent distribution. Moreover, we discuss a q-analogue for a result on non-overlapping segmented POPs. Finally, we suggest several open problems for further research.

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