Abstract
We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers:
(1 - E(1)x + E-3 x(3)/3! - E-4 x(4)/4! + E-6 x(6)/6! - E-7 x(7)/7! + ...)(-1), (*)
where Sigma(infinity)(n=0) E(n)x(n)/n! = sec x + tan x. We give two proofs of this formula. The first uses a system of differential equations whose solution gives the generating function
3 sin (1/2x) + 3 cosh (1/2 root 3x)/3 cos (1/2x) - root 3 sinh (1/2 root 3x),
which we then show is equal to (*). The second proof derives (*) directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function (*) is an "alternating" analogue of David and Barton's generating function
(1 - x + x(3)/3! - x(4)/4! + x(6)/6! - x(7)/7! + ...)(-1),
for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel.