|
| |
|
|
A007999
|
|
a(n) is the number of permutations w of 1,2,...,n such that both w and w^{-1} are alternating.
|
|
3
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
| |
|
|
