A Survey of Alternating Permutations

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 André that if En is the number of alternating permutations of 1, 2, . . . , n, then P n≥0 En x n n! = 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 1. Basic enumerative properties. Let Sn denote the symmetric group of all permutations of [n] := {1, 2, . . . , n}. A permutation w = a1a2 · · · an ∈ Sn is called alternating if a1 > a2 < a3 > a4 < · · · . In other words, ai < ai+1 for i even, and ai > ai+1 for i odd. Similarly w is reverse alternating if a1 < a2 > a3 < a4 > · · · . (Some authors reverse these definitions.) Let En denote the number of alternating permutations in Sn. (Set E0 = 1.) For instance, E4 = 5, corresponding to the permutations 2143, 3142, 3241, 4132, and 4231. The number En is called an Euler number because Euler considered the numbers E2n+1, though not with the combinatorial definition just given. (Rather, Euler defined them via equation (1.3) below.) The involution (1.1) a1a2 · · · an 7→ n+ 1− a1, n+ 1− a2, · · · , n+ 1− an on Sn shows that En is also the number of reverse alternating permutations in Sn. We write Altn (respectively, Raltn) for the set of alternating (respectively, reverse alternating) permutations w ∈ Sn. The subject of alternating permutations and Euler numbers has become so vast that it is impossible to give a comprehensive survey. We will confine ourselves to some highlights and to some special topics that we find especially interesting. 2000 Mathematics Subject Classification. Primary 05E10, Secondary 05E05. This material is based upon work supported by the National Science Foundation under Grant No. 0604423. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect those of the National Science Foundation. 2 RICHARD P. STANLEY The fundamental enumerative property of alternating permutations is due to Desiré André [1] in 1879. (Note however that Ginsburg [32] asserts without giving a reference that Binet was aware before André that the coefficients of secx count alternating permutations.) Theorem 1.1. We have