A053501 Number of degree-n permutations of order dividing 11.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3628801, 43545601, 283046401, 1320883201, 4953312001, 15850598401, 44910028801, 115482931201, 274271961601, 609493248001, 1279935820801, 4644633666390681601, 106826520356358566401, 1281918194457262387201

Offset: 0

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
• L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Crossrefs

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=11 of A008307.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^11/11) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

Maple

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 11])))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 14 2013

Mathematica

a[n_]:= n!*Sum[If[Mod[11*k-n, 10] == 0, Binomial[k, (11*k-n)/10]*11^((k-n)/10)/k!, 0], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 20 2014, after Vladimir Kruchinin *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^11/11], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

Maxima

a(n):=n!*sum(if mod(11*k-n,10)=0 then binomial(k,(11*k-n)/10)*(11)^((k-n)/10)/k! else 0,k,1,n); /* Vladimir Kruchinin, Sep 10 2010 */

PARI

my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^11/11) )) \\ G. C. Greubel, May 15 2019

Sage

m = 30; T = taylor(exp(x +x^11/11), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

Formula

E.g.f.: exp(x + x^11/11).

a(n) = n!*Sum_{k=1..n} (if mod(11*k-n,10)=0 then C(k,(11*k-n)/10)*(11)^((k-n)/10)/k!, else 0), n>0. - Vladimir Kruchinin, Sep 10 2010