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On the Limiting Distribution for the Longest Alternating Sequence in a Random Permutation

Harold Widom
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Abstract

Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the generating function (on n and k) for the probability that as(w) is at most k and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function.

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Harold Widom. 2005-11-21. On the Limiting Distribution for the Longest Alternating Sequence in a Random Permutation. https://arxiv.org/abs/math/0511533

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