A332344 -id:A332344 - cp's OEIS frontend

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

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

nonn

			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.

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]

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).

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 *)

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

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

A332345 a(n) is the number of totally alternating permutations of 1,2,...,n.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 28, 76, 240, 852, 3392, 14868, 71392, 371740, 2089184, 12589980, 81037792, 554555300, 4021728992, 30800978980, 248464000480, 2105271605804, 18696216008416, 173630895701996, 1683187452989920, 17000599430003444, 178625854452674272

Offset: 0

David Bevan, Feb 10 2020

nonn

A permutation w of 1,...,n is totally alternating if both w and w^{-1} are either alternating or reverse alternating.
Totally alternating permutations are those that avoid the consecutive patterns 123 and 321 and also the consecutive covincular patterns 123 and 321.
If a permutation is totally alternating, then so are its other 7 symmetries.

			The six totally alternating permutations of 1,...,4 are 1324, 2143, 2413, 3142, 3412 and 4231.

R. P. Stanley, Alternating permutations and symmetric functions, arXiv:math/0603520 [math.CO], 2006.

For n > 1, a(n) = 2*(A007999(n) + A332344(n)).
For odd n > 1, a(n) = 4*A007999(n).
For even n > 1, a(n) = 4*A007999(n) - 2*A007999(n-2).

cp's OEIS Frontend

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

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