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Alternating permutations

Richard Stanley

Friday, May 4, 2007
006 Kemeny Hall, 2 pm
Tea 3:30 pm, 300 Kemeny Hall

Abstract: A permutation a_1,a_2,\dots,a_n of 1,2,\dots,n is \emph{alternating} if a_1>a_2a_4<\cdots. The number of alternating permutations of 1,2,\dots,n is denoted E_n and satisfies

\sum_{n\geq 0}E_n\frac{x^n}{n!} =\sec x +\tan x.

After a survey of the basic properties of alternating permutations and the subject of ``combinatorial trigonometry,'' we will discuss recent work in two areas : (a) distribution of the length of the longest alternating subsequence of a permutation of 1,2,\dots,n, and (b) enumeration of various classes of alternating permutations of 1,2,\dots,n (such as those that are involutions) using techniques from symmetric functions.

This talk will be accessible to graduate students.