Clusters, generating functions and asymptotics for consecutive patterns in permutations

Elizalde, Sergi; Noy, Marc

doi:10.1016/j.aam.2012.08.003

Cited by 31 publications

(65 citation statements)

References 18 publications

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“…It is therefore surprising that the behavior of these two patterns is completely different in the consecutive setting, where they are the most and the least avoided patterns, respectively.The third result in this paper concerns non-overlapping patterns. We prove a recent conjecture of Elizalde and Noy [17] stating that among non-overlapping patterns of length…”

mentioning

confidence: 63%

“…The third result in this paper concerns non-overlapping patterns. We prove a recent conjecture of Elizalde and Noy [17] stating that among non-overlapping patterns of length…”

mentioning

confidence: 63%

“…In this section, we study non-overlapping patterns with two purposes. On one hand, we prove Conjecture 7.1 from [17], stating that among non-overlapping patterns of length m, the pattern 134 . .…”

Section: Non-overlapping Patternsmentioning

confidence: 95%

“…The computation of the generating functions P σ (u, z) is simplified by using an adaptation of the cluster method of Goulden and Jackson [20,21], which is based on inclusion-exclusion. We now summarize this adaptation to consecutive patterns in permutations, which has been recently used in [10,17,22]. For fixed σ ∈ S m , a k-cluster of length n with respect to σ is a pair (π; i 1 , i 2 , .…”

Section: The Cluster Methodsmentioning

confidence: 99%

“…The case m = 3 was proved in [16]. More recently, Elizalde and Noy [17] showed that the number of permutations avoiding 12 . .…”

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confidence: 97%

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The most and the least avoided consecutive patterns

Elizalde

2012

Proceedings of the London Mathematical Society

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We prove that the number of permutations avoiding the consecutive pattern 12… m, that is, containing no m adjacent entries in increasing order, is asymptotically larger than the number of permutations avoiding any other consecutive pattern of length m. This settles a conjecture of Elizalde and Noy from 2001. We also prove a recent conjecture of Nakamura stating that, at the other end of the spectrum, the number of permutations avoiding 12… (m−2)m(m−1) is asymptotically smaller than for any other pattern. Finally, we consider non‐overlapping patterns and obtain analogous results describing the most and least avoided ones. The techniques used include the cluster method of Goulden and Jackson, an interpretation of clusters as linear extensions of posets, and singularity analysis of generating functions.

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confidence: 63%

“…The third result in this paper concerns non-overlapping patterns. We prove a recent conjecture of Elizalde and Noy [17] stating that among non-overlapping patterns of length…”

mentioning

confidence: 63%

Section: Non-overlapping Patternsmentioning

confidence: 95%

Section: The Cluster Methodsmentioning

confidence: 99%

“…The case m = 3 was proved in [16]. More recently, Elizalde and Noy [17] showed that the number of permutations avoiding 12 . .…”

mentioning

confidence: 97%

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The most and the least avoided consecutive patterns

Elizalde

2012

Proceedings of the London Mathematical Society

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The probability of avoiding consecutive patterns in the Mallows distribution

Crane

DeSalvo

Elizalde

2018

Random Struct Algorithms

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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. K E Y W O R D S consecutive pattern, inversion, Mallows distribution, permutation, Stein's method Let S n be the set of permutations of [n] = {1, . . . , n}. Given a list of m distinct integers w = w 1 . . . w m , the standardization of w, written st(w), is the unique permutation of [m] that is order-isomorphic to Random Struct Alg. 2018;00:1-31.wileyonlinelibrary.com/journal/rsa

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Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one‐dimensional labels

Borga

2021

Random Struct Algorithms

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We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one‐dimensional‐labeled generating tree (together with some technical assumptions) and implies local convergence for random permutations in such families. We exhibit ten different families of permutations, most of them being permutation classes, that satisfy our assumptions. To the best of our knowledge, this is the first work where generating trees—which were introduced to enumerate combinatorial objects—have been used to establish probabilistic results.

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Clusters, generating functions and asymptotics for consecutive patterns in permutations

Cited by 31 publications

References 18 publications

The most and the least avoided consecutive patterns

The most and the least avoided consecutive patterns

The probability of avoiding consecutive patterns in the Mallows distribution

Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one‐dimensional labels

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