ABSTRACT |
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. |