A007999 a(n) is the number of permutations w of 1,2,...,n such that both w and w^{-1} are alternating.

1, 1, 1, 1, 2, 3, 8, 19, 64, 213, 880, 3717, 18288, 92935, 531440, 3147495, 20525168, 138638825, 1015694832, 7700244745, 62623847536, 526317901451, 4705365925872, 43407723925499, 423149546210416, 4250149857500861, 44868038386273776, 487341646372204813

Offset: 0

Author

poirier(AT)lacim.uqam.ca, Simon Plouffe

Keywords

• nonn

Examples

			The only alternating permutation of 1,2,3 whose inverse is alternating is 132.
The two alternating permutations of 1,2,3,4 whose inverses are alternating are 1324 and 3412.
		

Links

• H. O. Foulkes, Tangent and secant numbers and representations of symmetric groups, Discrete Math. 15 (1976), no. 4, 311-324.
• R. P. Stanley, Alternating permutations and symmetric functions, arXiv:math/0603520 [math.CO], 2006. [_Joel B. Lewis_, May 21 2009]

Crossrefs

Cf. A000111.

For odd n, a(n) = A332344(n).

For even n > 1, a(n) - a(n-2) = A332344(n).

For n > 1, a(n) = A332345(n)/2 - A332344(n).

Programs

Mathematica

m = 27;
e[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1)(2^(n+1)-1)*BernoulliB[ n+1])/(n+1)]];
u[x_] := Log[(1+x)/(1-x)]/2;
Sum[e[2k+1]^2 u[x]^(2k+1)/(2k+1)!, {k, 0, m}] + (1-x^2)^(-1/2) Sum[e[2k]^2* u[x]^(2k)/(2k)!, {k, 0, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Feb 24 2019 *)

Formula

G.f.: Sum_{k>=0} E_{2k+1}^2 u^(2k+1)/(2k+1)! + (1-x^2)^(-1/2) Sum_{k>=0} E_{2k}^2 u^(2k)/(2k)!, where E_j is an Euler number (A000111) and u = (1/2)*log((1+x)/(1-x)). - Richard Stanley, Jan 21 2006

Extensions

More terms from Vladeta Jovovic, May 15 2007

Two initial terms (thus correcting first term index, and consequent correction of Mathematica code) added by David Bevan, Feb 10 2020