Introduction to partially ordered patterns

Research output: Contribution to journalArticle

7 Citations (Scopus)

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.
LanguageEnglish
Pages929-944
Number of pages16
JournalDiscrete Applied Mathematics
Volume155
Issue number8
Early online date27 Sep 2006
DOIs
Publication statusPublished - 15 Apr 2007

Fingerprint

Permutation
Narayana numbers
Pell numbers
Catalan number
Shuffle
Q-analogue
Descent
Poset
Generating Function
Open Problems
Corollary
Arbitrary
Generalization
Review
Class

Keywords

  • non-overlapping occurrences of patterns
  • q-analogue
  • flat poset
  • co-unimodal pattern
  • bijection
  • generating function
  • Catalan numbers
  • Narayana numbers
  • Pell numbers

Cite this

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title = "Introduction to partially ordered patterns",
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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Introduction to partially ordered patterns. / Kitaev, Sergey.

In: Discrete Applied Mathematics, Vol. 155, No. 8, 15.04.2007, p. 929-944.

Research output: Contribution to journalArticle

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AB - 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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