A237036 - OEIS

%I #30 Apr 07 2018 07:42:15

%S 3,12,40,70,105,168,240,330,440,572,728,910,1120,1360,1632,1938,2280,

%T 2660,3080,3542,4048,4600,5200,5850,6552,7308,8120,8990,9920,10912,

%U 11968,13090,14280,15540,16872,18278,19760,21320,22960,24682,26488,28380,30360,32430

%N Size of the smallest conjugacy class of size greater than 1 of the alternating group of degree n.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Alternating_group">Alternating group</a>

%F From _Alois P. Heinz_, Feb 04 2014: (Start)

%F G.f.: -x^4*(7*x^8-28*x^7+42*x^6-20*x^5-20*x^4+30*x^3-10*x^2-3)/(x-1)^4.

%F a(n) = 2*C(n,3) = A007290(n) for n>=9. (End)

%e For n = 4 the conjugacy classes of size greater than 1 of Alt(n) are

%e {(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)},

%e {(2,4,3), (1,2,3), (1,3,4), (1,4,2)},

%e {(2,3,4), (1,2,4), (1,3,2), (1,4,3)},

%e the smallest of which has 3 elements, hence a(4) = 3.

%t Join[{3,12,40,70,105},2*Binomial[Range[9,50],3]] (* _Harvey P. Dale_, Apr 07 2018 *)

%o (GAP)

%o a:=function(n)

%o local G,CC,SCC,SCC1;

%o G:=AlternatingGroup(n);

%o CC:=ConjugacyClasses(G);;

%o SCC:=List(CC,Size);

%o SCC1:=Difference(SCC,[1]);

%o return Minimum(SCC1);

%o end;;

%Y Cf. A000702, A070733, A007290.

%K nonn

%O 4,1

%A _W. Edwin Clark_, Feb 02 2014