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FIELD Colloquium August 12 (Mon), 2013 11:00-13:00 1503 Yoo, HwanChul Massachusetts Institute of Technology [Colloquium]Alternating permutations A permutation $a_1,a_2,\dots,a_n$ of $1,2,\dots,n$ is called \emph{alternating} if $a_1>a_2a_4<\cdots$. The number of alternating permutations of $1,2,\dots,n$ is denoted $E_n$ and is called an \emph{Euler number}. The most striking result about alternating permutations is the generating function $$\sum_{n\geq 0}E_n\frac{x^n}{n!} = \sec x+\tan x,$$ found by D\'esir\'e Andr\'e in 1879. We will discuss this result and how it leads to the subject of combinatorial trigonometry.'' We will then survey some further aspects of alternating permutations, including some other objects that are counted by $E_n$, the use of the representation theory of the symmetric group to count certain classes of alternating permutations, and the distribution of the length of longest alternating subsequence of a random permutation.

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