### 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) |
---|---|

Pages (from-to) | 417-447 |

Number of pages | 31 |

Journal | Random Structures and Algorithms |

Volume | 53 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 2018 |

### Fingerprint

### 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

### Cite this

*Random Structures and Algorithms*,

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

}

*Random Structures and Algorithms*, vol. 53, no. 3, pp. 417-447. https://doi.org/10.1002/rsa.20776

**The probability of avoiding consecutive patterns in the Mallows distribution.** / Crane, Harry; DeSalvo, Stephen; Elizalde, Sergi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The probability of avoiding consecutive patterns in the Mallows distribution

AU - Crane, Harry

AU - DeSalvo, Stephen

AU - Elizalde, Sergi

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - Mallows distribution

KW - Stein's method

KW - consecutive pattern

KW - inversion

KW - permutation

UR - http://www.scopus.com/inward/record.url?scp=85044775127&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044775127&partnerID=8YFLogxK

U2 - 10.1002/rsa.20776

DO - 10.1002/rsa.20776

M3 - Article

AN - SCOPUS:85044775127

VL - 53

SP - 417

EP - 447

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -