# The probability of avoiding consecutive patterns in the Mallows distribution

Harry Crane, Stephen DeSalvo, Sergi Elizalde

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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) 417-447 31 Random Structures and Algorithms 53 3 https://doi.org/10.1002/rsa.20776 Published - Oct 1 2018

### Fingerprint

Consecutive
Normal distribution
Inversion
Permutation
Stein's Method
Singularity Analysis
Random Permutation
Q-analogue
Uniform distribution
Weighting
Gaussian distribution
Monotone
Generalise

### 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

Crane, Harry ; DeSalvo, Stephen ; Elizalde, Sergi. / The probability of avoiding consecutive patterns in the Mallows distribution. In: Random Structures and Algorithms. 2018 ; Vol. 53, No. 3. pp. 417-447.
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The probability of avoiding consecutive patterns in the Mallows distribution. / Crane, Harry; DeSalvo, Stephen; Elizalde, Sergi.

In: Random Structures and Algorithms, Vol. 53, No. 3, 01.10.2018, p. 417-447.

Research output: Contribution to journalArticle

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AU - DeSalvo, Stephen

AU - Elizalde, Sergi

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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.

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