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
Keywords
Examples
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.
Links
- Wikipedia, Alternating group
Programs
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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;;
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Mathematica
Join[{3,12,40,70,105},2*Binomial[Range[9,50],3]] (* Harvey P. Dale, Apr 07 2018 *)
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)