This survey of alternating permutations and Euler numbers includes
refinements of Euler numbers, other occurrences of Euler numbers, longest
alternating subsequences, umbral enumeration of classes of alternating
permutations, and the cd-index of the symmetric group.Comment: 32 pages, 7 figure

Stanley, Richard P.

English

arXiv

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

A Survey of Alternating Permutations

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