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A survey of alternating permutations

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Friday, September 30, 2016, 1:05 pm– 2 pm

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Speaker: Professor Richard Stanley, MIT


A permutation a1 a2 ··· an is alternating if a1 > a2 < a3 > a4 < a5 > ···. If En is the number of alternating permutations of 1, 2, ..., n, then Σn ≥ 0 En xn (n!)-1 = sec x + tan x. We will discuss several aspects of the theory of alternating permutations. Some occurences of the numbers En, such as counting orbits of group actions and volumes of polytopes, will be surveyed. The behavior of the length of the longest alternating subsequence of a random permutation will be analyzed, in analogy to the length of the longest increasing subsequence. We will also explain how various classes of alternating permutations, such as those that are also fixed-point free involutions, can by counted using a certain representation of the symmetric group Sn whose dimension is En.

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