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## Alternating permutations

### Richard Stanley

MIT

###
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.