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

0,5

Links

Table of n, a(n) for n=0..27.

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]

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

Example

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.

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 *)

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).

Sequence in context: A243791 A243335 A303835 * A006609 A005663 A112834

Adjacent sequences: A007996 A007997 A007998 * A008000 A008001 A008002

Keyword

nonn

Author

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

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

Status

approved