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[Video] A Survey of Alternating Permutations

Richard Stanley, Massachusetts Institute of Technology
Friday, January 14, 2011 - 2:30pm

An alternating permutation \(w=a_1\cdots a_n\) of \(1,2,\dots,n\) 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 \(1,2,\dots,n\), then \(\sum_{n\geq 0}E_n\frac{x^n}{n!}=\sec x+\tan x\). We will discuss such topics as other occurrences of Euler numbers in mathematics, umbral enumeration of classes of alternating permutations, and longest alternating subsequences of permutations.

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