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%I #6 Dec 05 2018 07:58:19
%S 0,0,0,0,0,1,0,1,1,2,0,4,0,4,3,5,0,9,0,10,5,11,1,18,3,17,8,22,3,35,5,
%T 32,17,39,16,59,14,58,33,75,28,103,35,106,71,125,63,174,81,192,127,
%U 220,130,294,170,325,237,378,257,504
%N Number of strict 2-edge-connected integer partitions of n.
%C An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>
%e The a(24) = 18 strict 2-edge-connected integer partitions of 24:
%e (15,9) (10,8,6) (10,8,4,2)
%e (16,8) (12,8,4) (12,6,4,2)
%e (18,6) (12,9,3)
%e (20,4) (14,6,4)
%e (21,3) (14,8,2)
%e (22,2) (15,6,3)
%e (14,10) (16,6,2)
%e (18,4,2)
%e (12,10,2)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
%t Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,twoedQ[primeMS/@#]]&]],{n,30}]
%Y Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.
%K nonn
%O 1,10
%A _Gus Wiseman_, Dec 04 2018
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