A Survey of Alternating Permutations
Author(s)Stanley, Richard P.
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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.
Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Combinatorics and Graphs
American Mathematical Society
Stanley, Richard P. "A Survey of Alternating Permutations." in Combinatorics and Graphs: the twentieth anniversary conference of IPM Combinatorics, May 15-21, 2009, Tehran, Iran, edited by Richard A. Brualdi, Samad Hedayat, Hadi Kharaghani, Gholamriza B. Khosrovshahi and Shahrair Shahrairi.(in Contemporary Mathematics), American Mathematical Society, 2010.
Author's final manuscript