cp's OEIS Frontend

A066052 Number of permutations in the symmetric group S_n with order >= 3.

%I A066052 #25 Jul 07 2020 03:24:09
%S A066052 0,0,2,14,94,644,4808,39556,360260,3619304,39881104,478861448,
%T A066052 6226452296,87175900720,1307664018464,20922743681264,355687216296688,
%U A066052 6402372708414176,121645095599130560,2432901984417975904,51090942051756747104,1124000727158723041088,25852016735627132757376
%N A066052 Number of permutations in the symmetric group S_n with order >= 3.
%C A066052 Permutations which contain a cycle with length >=3.
%C A066052 For elements of order >= 3 we can show that they can be represented as a product of two involutions. [Proof: Write down the reduced cycle representation of the element. Decompose each sub-cycle of order >=3 into a product of two involutions, and distribute/merge the 1 or 2 factors of the involutions. For sub-cycles of order o>=3 it suffices to show that the reduced cycle (1 2 3 4 ... o), equivalent to the 1-line representation (2 3 4 .. o 1), can be written as a product of 2 involutions, because other cycles are mapped to such a product by a fixed permutation. A constructive proof here is: the reduced cycle (1 2 3 4 ... o) is the product of the two involutions (1 2)(3 o)(4 o-1)(5 o-2)... and (2 o)(3 o-1)... The first involution swaps the first two elements and reverts the ordering of all others from the 3rd on, then the second involution reverts the order of the elements from the 2nd on. The net effect is a cyclic shift of the o elements.] - _R. J. Mathar_, Jan 04 2017
%C A066052 a(n) is the number of permutations of [n] that are not involutions (see first formula). - _Joerg Arndt_, Jul 06 2020
%F A066052 a(n) = n! - A000085(n).
%F A066052 E.g.f.: 1/(1-x) - exp(x*(x+2)/2).
%F A066052 Conjecture: (-n+3)*a(n) +(n^2-2*n-1)*a(n-1) +(-n+1)*a(n-2) -(n-1)*(n-2)^2*a(n-3)=0. - _R. J. Mathar_, Jan 04 2017
%F A066052 Conjecture: a(n) +(-n-3)*a(n-1) +(2*n-1)*a(n-2) +n*(n-2)*a(n-3) -2*(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 04 2017
%t A066052 nn=20; Range[0,nn]! CoefficientList[Series[1/(1-x)-Exp[x+x^2/2],{x,0,nn}],x]  (* _Geoffrey Critzer_, Jan 09 2013 *)
%Y A066052 Cf. A000085, A000142.
%K A066052 nonn,easy
%O A066052 1,3
%A A066052 Sharon Sela (sharonsela(AT)hotmail.com), Dec 29 2001
%E A066052 More terms from _Vladeta Jovovic_, Dec 31 2001