A Survey of Alternating Permutations

Abstract

A permutation a1a2 · · · an of 1, 2, . . . , n is alternating if a1 > a2 < a3 > a4 < · · · . We survey some aspects of the theory of alternating permutations, beginning with the famous result of Andr´e that if En is the number of alternating permutations of 1, 2, . . . , n, then ... = sec x + tan x. Topics include refinements and q-analogues of En, various occurrences of En in mathematics, longest alternating subsequences of permutations, umbral enumeration of special classes of alternating permutations, and the connection between alternating permutations and the cd-index of the symmetric group.