A237036 Size of the smallest conjugacy class of size greater than 1 of the alternating group of degree n.

3, 12, 40, 70, 105, 168, 240, 330, 440, 572, 728, 910, 1120, 1360, 1632, 1938, 2280, 2660, 3080, 3542, 4048, 4600, 5200, 5850, 6552, 7308, 8120, 8990, 9920, 10912, 11968, 13090, 14280, 15540, 16872, 18278, 19760, 21320, 22960, 24682, 26488, 28380, 30360, 32430

Offset

4,1

Links

Table of n, a(n) for n=4..47.

Wikipedia, Alternating group

Formula

From Alois P. Heinz, Feb 04 2014: (Start)

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.

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

Example

For n = 4 the conjugacy classes of size greater than 1 of Alt(n) are
{(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)},
{(2,4,3), (1,2,3), (1,3,4), (1,4,2)},
{(2,3,4), (1,2,4), (1,3,2), (1,4,3)},
the smallest of which has 3 elements, hence a(4) = 3.

Mathematica

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

Prog

(GAP)
a:=function(n)
local G, CC, SCC, SCC1;
G:=AlternatingGroup(n);
CC:=ConjugacyClasses(G);;
SCC:=List(CC, Size);
SCC1:=Difference(SCC, [1]);
return Minimum(SCC1);
end;;

Crossrefs

Cf. A000702, A070733, A007290.

Sequence in context: A271218 A062311 A303348 * A034956 A032093 A007993

Adjacent sequences: A237033 A237034 A237035 * A237037 A237038 A237039

Keyword

nonn

Author

W. Edwin Clark, Feb 02 2014

Status

approved