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A237036
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Size of the smallest conjugacy class of size greater than 1 of the alternating group of degree n.
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0
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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
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OFFSET
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4,1
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LINKS
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FORMULA
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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)
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EXAMPLE
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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.
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MATHEMATICA
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Join[{3, 12, 40, 70, 105}, 2*Binomial[Range[9, 50], 3]] (* Harvey P. Dale, Apr 07 2018 *)
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PROG
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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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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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