### Abstract

We use combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution is a q-analogue of the uniform distribution weighting each permutation π by q^{inv(π)}, where inv(π) is the number of inversions in π and q is a positive, real-valued parameter. We prove that the growth rate exists for all patterns and all q > 0, and we generalize Goulden and Jackson's cluster method to keep track of the number of inversions in permutations avoiding a given consecutive pattern. Using singularity analysis, we approximate the growth rates for length-3 patterns, monotone patterns, and non-overlapping patterns starting with 1, and we compare growth rates between different patterns. We also use Stein's method to show that, under certain assumptions on q and σ, the number of occurrences of a given pattern σ is well approximated by the normal distribution.

Original language | English (US) |
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Pages (from-to) | 417-447 |

Number of pages | 31 |

Journal | Random Structures and Algorithms |

Volume | 53 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2018 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

### Keywords

- Mallows distribution
- Stein's method
- consecutive pattern
- inversion
- permutation

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## Cite this

*Random Structures and Algorithms*,

*53*(3), 417-447. https://doi.org/10.1002/rsa.20776