A137482 - OEIS

%I #19 Feb 19 2019 20:06:30

%S 1,1,0,2,14,54,304,2260,18108,161756,1618496,17815896,213767080,

%T 2778833992,38904145344,583563781424,9337011390224,158729175524880,

%U 2857125341582848,54285381652008736,1085707629235539936,22799860214350346336,501596924799005576960

%N Number of permutations of n objects such that no two-element subset is preserved.

%C In other words, there are no two objects which the permutation leaves fixed and no two objects that the permutation swaps.

%C The limit as n -> infinity of a(n)/n! = 2/exp(3/2) or approximately 0.4462603203. - _Les Reid_, Jun 04 2012

%H Alois P. Heinz, <a href="/A137482/b137482.txt">Table of n, a(n) for n = 0..450</a>

%H Hannah Jackson, Kathryn Nyman and Les Reid, <a href="http://dx.doi.org/10.2140/involve.2013.6.25">Properties of generalized derangement graphs</a>, Involve, Vol. 6 (2013), No. 1, 25-33; DOI: 10.2140/involve.2013.6.25.

%F E.g.f.: (1+x)*exp(-x)*exp(-x^2/2)/(1-x).

%F a(n) = (n-1)*a(n-1) - a(n-2) + (n-2)*n*a(n-3) for n > 2, a(n) = (n+1)*(2-n)/2 for n < 3. - _Alois P. Heinz_, Feb 19 2019

%e a(3)=2 because we have 312 and 231.

%p g:=(1+x)*exp(-x)*exp(-(1/2)*x^2)/(1-x): gser:=series(g,x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<3, (n+1)*(2-n)/2,

%p       (n-1)*a(n-1)-a(n-2)+(n-2)*n*a(n-3))

%p     end:

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Feb 19 2019

%t With[{nn=20},CoefficientList[Series[((1+x)Exp[-x]Exp[-x^2/2])/(1-x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 17 2013 *)

%Y Cf. A000166, A000266.

%K nonn

%O 0,4

%A Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008, corrected Apr 30 2008