cp's OEIS Frontend

A218970 Number of connected cyclic conjugacy classes of subgroups of the symmetric group.

%I A218970 #17 Dec 05 2018 07:57:54
%S A218970 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 A218970 144,112,203,152,301,219,393,342,540,474,770,661,1022,967,1397,1313,
%U A218970 1928,1821,2565,2564,3439,3445,4676,4687,6186,6406,8215,8543,10974,11435
%N A218970 Number of connected cyclic conjugacy classes of subgroups of the symmetric group.
%C A218970 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 A218970 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 A218970 Liam Naughton, <a href="http://www.maths.nuigalway.ie/~liam/CountingSubgroups.g">CountingSubgroups.g</a>
%H A218970 Liam Naughton and Goetz Pfeiffer, <a href="http://schmidt.nuigalway.ie/tomlib/">Tomlib, The GAP table of marks library</a>
%F A218970 For n > 1, a(n) = A304716(n) - 1. - _Gus Wiseman_, Dec 03 2018
%e A218970 From _Gus Wiseman_, Dec 03 2018: (Start)
%e A218970 The a(12) = 14 connected integer partitions of 12:
%e A218970   (12)  (6,6)   (4,4,4)  (3,3,3,3)  (4,2,2,2,2)  (2,2,2,2,2,2)
%e A218970         (8,4)   (6,3,3)  (4,4,2,2)
%e A218970         (9,3)   (6,4,2)  (6,2,2,2)
%e A218970         (10,2)  (8,2,2)
%e A218970 (End)
%t A218970 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 A218970 Table[Length[Select[IntegerPartitions[n],Length[zsm[#]]==1&]],{n,10}]
%Y A218970 Cf. A018783, A200976, A286518, A286520, A290103, A304714, A304716, A305078, A305079, A322306, A322307.
%K A218970 nonn
%O A218970 0,5
%A A218970 _Liam Naughton_, Nov 26 2012
%E A218970 More terms from _Gus Wiseman_, Dec 03 2018