Permutations avoiding consecutive patterns, II

We consider the problem of enumerating permutations in the symmetric group on n elements which avoid a given set of consecutive patterns S, and in particular computing asymptotics as n tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators onL2([0,1]m), where the patterns in S have length m+1.Kre\u{\i}n and Rutman’s generalization of the Perron–Frobenius theory of non-negative matrices plays a central role. Our methods give detailed asymptotic expansions and allow for explicit computation of leading terms in many cases.As a corollary to our results,we settle a conjecture of Warlimont on asymptotics for the number of permutations avoiding a consecutive pattern

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“…As a corollary we have the following result, proving a conjecture of Warlimont [23]. See also Theorem 4.1 in [9].…”

Section: Introductionsupporting

confidence: 52%

A spectral approach to consecutive pattern-avoiding permutations

Kitaev

Ehrenborg

Perry

2011

Journal of Combinatorics

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“…Compare this result with the conjecture of Warlimont [20] that for any consecutive pattern σ there exist constants γ > 0 and w < 1 such that α n (σ )/n! ∼ γ w n .…”

Section: Asymptotic Enumerationmentioning

confidence: 69%

Asymptotic enumeration of permutations avoiding generalized patterns

Elizalde

2006

Advances in Applied Mathematics

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Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them.We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns.

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“…To conclude, let us mention a recent result of Ehrenborg, Kitaev and Perry [5], proved using methods from spectral theory, and previously conjectured by Warlimont [19]. It would be interesting to find a more combinatorial proof of this important result using singularity analysis of generating functions.…”

Section: Open Problemsmentioning

confidence: 83%

Clusters, generating functions and asymptotics for consecutive patterns in permutations

Elizalde

Noy

2012

Advances in Applied Mathematics

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We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function.We also complete the classification of consecutive patterns of length up to 6 into equivalence classes, proving a conjecture of Nakamura. Finally, we show that the monotone pattern asymptotically dominates (in the sense that it is easiest to avoid) all non-overlapping patterns of the same length, thus proving a conjecture of Elizalde and Noy for a positive fraction of all patterns.

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Permutations avoiding consecutive patterns, II

Abstract: The generating power series for the number of permutations avoiding particular consecutive patterns are derived in a new and simple fashion.

Cited by 14 publications

References 2 publications

A spectral approach to consecutive pattern-avoiding permutations

A spectral approach to consecutive pattern-avoiding permutations

Asymptotic enumeration of permutations avoiding generalized patterns

Clusters, generating functions and asymptotics for consecutive patterns in permutations

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