Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns

Anders Claesson, Vít Jelínek, Einar Steingrimsson

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

We prove that the Stanley-Wilf limit of any layered permutation pattern of length l is at most 4l(2), and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern.

We also conjecture that, for any k >= 0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n + 1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most e(pi root 2/3) similar or equal to 13.001954.
Original languageEnglish
Pages (from-to)1680-1691
Number of pages12
JournalJournal of Combinatorial Theory Series A
Volume119
Issue number8
DOIs
Publication statusPublished - Nov 2012

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Keywords

  • upper bounds
  • stanley-wilf limit
  • layered patterns
  • pattern avoidance
  • layered permutations

Cite this

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abstract = "We prove that the Stanley-Wilf limit of any layered permutation pattern of length l is at most 4l(2), and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. We also conjecture that, for any k >= 0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n + 1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most e(pi root 2/3) similar or equal to 13.001954.",
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Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns. / Claesson, Anders; Jelínek, Vít; Steingrimsson, Einar.

In: Journal of Combinatorial Theory Series A , Vol. 119, No. 8, 11.2012, p. 1680-1691.

Research output: Contribution to journalArticle

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AU - Steingrimsson, Einar

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AB - We prove that the Stanley-Wilf limit of any layered permutation pattern of length l is at most 4l(2), and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. We also conjecture that, for any k >= 0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n + 1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most e(pi root 2/3) similar or equal to 13.001954.

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