A Survey of Alternating Permutations

• Main Author: Stanley, Richard P. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: American Mathematical Society, 2012-07-26T19:06:45Z.
• Subjects: Article
• Online Access: Get fulltext

Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday.
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.
National Science Foundation (U.S.) (grant No. 0604423)