The most and the least avoided consecutive patterns

Elizalde, Sergi

doi:10.1112/plms/pds063

Cited by 17 publications

(44 citation statements)

References 24 publications

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“…In particular, analyzing their structure in the case of non-overlapping patterns will lead to the proof of the following result, which appears in Section 3. The above theorem states that if the conjecture from [7] about non-overlapping patterns holds, then so does our more general conjecture about arbitrary patterns, and thus these two conjectures are equivalent.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 88%

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Wilf equivalence relations for consecutive patterns

Dwyer

Elizalde

2018

Advances in Applied Mathematics

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Two permutations π and τ are c-Wilf equivalent if, for each n, the number of permutations in S n avoiding π as a consecutive pattern (i.e., in adjacent positions) is the same as the number of those avoiding τ . In addition, π and τ are strongly c-Wilf equivalent if, for each n and k, the number of permutations in S n containing k occurrences of π as a consecutive pattern is the same as for τ . In this paper we introduce a third, more restrictive equivalence relation, defining π and τ to be super-strongly c-Wilf equivalent if the above condition holds for any set of prescribed positions for the k occurrences. We show that, when restricted to non-overlapping permutations, these three equivalence relations coincide.We also give a necessary condition for two permutations to be strongly c-Wilf equivalent. Specifically, we show that if π, τ ∈ S m are strongly c-Wilf equivalent, then |π m − π 1 | = |τ m − τ 1 |.In the special case of non-overlapping permutations π and τ , this proves a weaker version of a conjecture of the second author stating that π and τ are c-Wilf equivalent if and only if π 1 = τ 1 and π m = τ m , up to trivial symmetries. Finally, we strengthen a recent result of Nakamura and Khoroshkin-Shapiro giving sufficient conditions for strong c-Wilf equivalence.

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Section: Introduction and Summary Of Resultsmentioning

confidence: 88%

“…3.2] and [14,Thm. 11] in the special case of non-overlapping permutations: 7,14]). Let π, τ ∈ S m be non-overlapping.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Wilf equivalence relations for consecutive patterns

Dwyer

Elizalde

2018

Advances in Applied Mathematics

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“…We remark that the natural analogue of this conjecture for ordinary Wilf-equivalence is already false for patterns of length 3 [6]. This conjecture has already been resolved in the case that π, τ are non-overlapping [8,14].…”

Section: Conjectures and Further Questionsmentioning

confidence: 83%

“…We now evaluate the the left-and right-hand sides of (8). The integral on the right-hand side of (8) is a multivariate beta (or Dirichlet) integral [4] and evaluates to…”

Section: Probabilistic Interpretation Of Linear Extensionsmentioning

confidence: 99%

“…Variations on this problem involve adding restrictions to the pattern: for instance, requiring that it appear in consecutive positions. This notion of consecutive pattern avoidance was first systematically studied by Elizalde and Noy in 2003 [10] and has been considered from multiple viewpoints [7,8,15]. For more information on the consecutive pattern avoidance literature, we refer the reader to the 2016 survey of Elizalde [9].…”

Section: Introductionmentioning

confidence: 99%

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Constraining strong c-Wilf equivalence using cluster poset asymptotics

Lee

Sah

2019

Advances in Applied Mathematics

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Let π ∈ Sm and σ ∈ Sn be permutations. An occurrence of π in σ as a consecutive pattern is a subsequence σiσi+1 · · · σi+m−1 of σ with the same order relations as π. We say that patterns π, τ ∈ Sm are strongly c-Wilf equivalent if for all n and k, the number of permutations in Sn with exactly k occurrences of π as a consecutive pattern is the same as for τ . In 2018, Dwyer and Elizalde [6] conjectured (generalizing a conjecture of Elizalde [8] from 2012) that if π, τ ∈ Sm are strongly c-Wilf equivalent, then (τ1, τm) is equal to one of (π1, πm), (πm, π1), (m+1−π1, m+1−πm), or (m + 1 − πm, m + 1 − π1). We prove this conjecture using the cluster method introduced by Goulden and Jackson in 1979 [12], which Dwyer and Elizalde previously applied [6] to prove that |π1 − πm| = |τ1 − τm|. A consequence of our result is the full classification of c-Wilf equivalence for a special class of permutations, the non-overlapping permutations. Our approach uses analytic methods to approximate the number of linear extensions of the "cluster posets" of Elizalde and Noy [11]. arXiv:1807.04921v1 [math.CO]

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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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The most and the least avoided consecutive patterns

Cited by 17 publications

References 24 publications

Wilf equivalence relations for consecutive patterns

Wilf equivalence relations for consecutive patterns

Constraining strong c-Wilf equivalence using cluster poset asymptotics

The probability of avoiding consecutive patterns in the Mallows distribution

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