### Abstract

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 language | English |
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Pages (from-to) | 1680-1691 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory Series A |

Volume | 119 |

Issue number | 8 |

DOIs | |

Publication status | Published - Nov 2012 |

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### Keywords

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

### Cite this

*Journal of Combinatorial Theory Series A*,

*119*(8), 1680-1691. https://doi.org/10.1016/j.jcta.2012.05.006

}

*Journal of Combinatorial Theory Series A*, vol. 119, no. 8, pp. 1680-1691. https://doi.org/10.1016/j.jcta.2012.05.006

**Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns.** / Claesson, Anders; Jelínek, Vít; Steingrimsson, Einar.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns

AU - Claesson, Anders

AU - Jelínek, Vít

AU - Steingrimsson, Einar

PY - 2012/11

Y1 - 2012/11

N2 - 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.

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.

KW - upper bounds

KW - stanley-wilf limit

KW - layered patterns

KW - pattern avoidance

KW - layered permutations

U2 - 10.1016/j.jcta.2012.05.006

DO - 10.1016/j.jcta.2012.05.006

M3 - Article

VL - 119

SP - 1680

EP - 1691

JO - Journal of Combinatorial Theory Series A

JF - Journal of Combinatorial Theory Series A

SN - 0097-3165

IS - 8

ER -