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%I #17 Dec 05 2018 07:57:54
%S 1,1,1,1,2,1,4,1,5,3,8,2,14,3,17,11,24,10,40,16,53,35,71,43,112,68,
%T 144,112,203,152,301,219,393,342,540,474,770,661,1022,967,1397,1313,
%U 1928,1821,2565,2564,3439,3445,4676,4687,6186,6406,8215,8543,10974,11435
%N Number of connected cyclic conjugacy classes of subgroups of the symmetric group.
%C a(n) is also the number of connected partitions of n in the following sense. Given a partition of n, the vertices are the parts of the partition and two vertices are connected if and only if their gcd is greater than 1. We call a partition connected if the graph is connected.
%H Liam Naughton and Goetz Pfeiffer, <a href="http://arxiv.org/abs/1211.1911">Integer sequences realized by the subgroup pattern of the symmetric group</a>, arXiv:1211.1911 [math.GR], 2012-2013.
%H Liam Naughton, <a href="http://www.maths.nuigalway.ie/~liam/CountingSubgroups.g">CountingSubgroups.g</a>
%H Liam Naughton and Goetz Pfeiffer, <a href="http://schmidt.nuigalway.ie/tomlib/">Tomlib, The GAP table of marks library</a>
%F For n > 1, a(n) = A304716(n) - 1. - _Gus Wiseman_, Dec 03 2018
%e From _Gus Wiseman_, Dec 03 2018: (Start)
%e The a(12) = 14 connected integer partitions of 12:
%e (12) (6,6) (4,4,4) (3,3,3,3) (4,2,2,2,2) (2,2,2,2,2,2)
%e (8,4) (6,3,3) (4,4,2,2)
%e (9,3) (6,4,2) (6,2,2,2)
%e (10,2) (8,2,2)
%e (End)
%t zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
%t Table[Length[Select[IntegerPartitions[n],Length[zsm[#]]==1&]],{n,10}]
%Y Cf. A018783, A200976, A286518, A286520, A290103, A304714, A304716, A305078, A305079, A322306, A322307.
%K nonn
%O 0,5
%A _Liam Naughton_, Nov 26 2012
%E More terms from _Gus Wiseman_, Dec 03 2018
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