Introduction to partially ordered patterns

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10 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)929-944
Number of pages16
JournalDiscrete Applied Mathematics
Issue number8
Early online date27 Sept 2006
Publication statusPublished - 15 Apr 2007


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


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