login
A326241
Number of degree-n even permutations of order dividing 12
2
1, 1, 1, 3, 12, 36, 216, 1296, 10368, 78912, 634896, 5572656, 51817536, 477672768, 8268884352, 101752505856, 1417554660096, 20985416983296, 344834432195328, 5096129755468032, 70148917686998016
OFFSET
0,4
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap 4, Problem 22).
LINKS
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.
FORMULA
E.g.f.: (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) + (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)).
EXAMPLE
For n=3 the a(3)=3 solutions are (1), (1, 2, 3), (1, 3, 2) (permutations in cyclic notation).
MAPLE
E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) + (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):
S:= series(E, x, 31):
seq(coeff(S, x, i)*i!, i=0..30); # Robert Israel, Jul 08 2019
MATHEMATICA
With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]+1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]
KEYWORD
easy,nonn
AUTHOR
STATUS
approved