# [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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