Mathematics Colloquia and Seminars

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An alternating permutation w=a₁ ... a_n of 1,2,...,n is a permutation such that a_i>a_i+1 if and only if i is odd. If E_n (called an Euler number) denotes the number of alternating permutations of 1,2,...,n, then ∑_{n ≥ 0} E_n^xn⁄_n!=sec x+tan x. We will discuss such topics as other occurrences of Euler numbers in mathematics, umbral enumeration of classes of alternating permutations, longest alternating subsequences of permutations, and a connection with a noncommutative polynomial called the cd-index of the symmetric group.