### Abstract

Language | English |
---|---|

Pages | 105–122 |

Number of pages | 18 |

Journal | Journal of the London Mathematical Society |

Volume | 92 |

Issue number | 1 |

Early online date | 16 Jun 2015 |

DOIs | |

Publication status | Published - 1 Aug 2015 |

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### Keywords

- permutations
- permutation classes
- Av(3124)
- Łukasiewicz paths

### Cite this

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*Journal of the London Mathematical Society*, vol. 92, no. 1, pp. 105–122. https://doi.org/10.1112/jlms/jdv020

**Permutations avoiding 1324 and patterns in Łukasiewicz paths.** / Bevan, David.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Permutations avoiding 1324 and patterns in Łukasiewicz paths

AU - Bevan, David

PY - 2015/8/1

Y1 - 2015/8/1

N2 - The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.

AB - The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.

KW - permutations

KW - permutation classes

KW - Av(3124)

KW - Łukasiewicz paths

U2 - 10.1112/jlms/jdv020

DO - 10.1112/jlms/jdv020

M3 - Article

VL - 92

SP - 105

EP - 122

JO - Journal of the London Mathematical Society

T2 - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 1

ER -