16,271 research outputs found

    On the Asymptotic Statistics of the Number of Occurrences of Multiple Permutation Patterns

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    We study statistical properties of the random variables Xσ(π)X_{\sigma}(\pi), the number of occurrences of the pattern σ\sigma in the permutation π\pi. We present two contrasting approaches to this problem: traditional probability theory and the ``less traditional'' computational approach. Through the perspective of the first one, we prove that for any pair of patterns σ\sigma and τ\tau, the random variables XσX_{\sigma} and XτX_{\tau} are jointly asymptotically normal (when the permutation is chosen from SnS_{n}). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normality (up to a point) and derive explicit formulas for quite a few moments and mixed moments empirically, yet rigorously. The computational approach can also be extended to the case where permutations are drawn from a set of pattern avoiders to produce many empirical moments and mixed moments. This data suggests that some random variables are not asymptotically normal in this setting.Comment: 18 page

    A Spectral Approach to Consecutive Pattern-Avoiding Permutations

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    We consider the problem of enumerating permutations in the symmetric group on nn elements which avoid a given set of consecutive pattern SS, and in particular computing asymptotics as nn tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators on L2([0,1]m)L^{2}([0,1]^{m}), where the patterns in SS has length m+1m+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.Comment: a reference is added; corrected typos; to appear in Journal of Combinatoric

    The Brownian limit of separable permutations

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    We study random uniform permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion

    Patterns in random permutations avoiding the pattern 132

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    We consider a random permutation drawn from the set of 132-avoiding permutations of length nn and show that the number of occurrences of another pattern σ\sigma has a limit distribution, after scaling by nλ(σ)/2n^{\lambda(\sigma)/2} where λ(σ)\lambda(\sigma) is the length of σ\sigma plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.Comment: 32 page
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