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A322337 Number of strict 2-edge-connected integer partitions of n.

%I A322337 #6 Dec 05 2018 07:58:19
%S A322337 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 A322337 32,17,39,16,59,14,58,33,75,28,103,35,106,71,125,63,174,81,192,127,
%U A322337 220,130,294,170,325,237,378,257,504
%N A322337 Number of strict 2-edge-connected integer partitions of n.
%C A322337 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 A322337 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>
%e A322337 The a(24) = 18 strict 2-edge-connected integer partitions of 24:
%e A322337   (15,9)   (10,8,6)   (10,8,4,2)
%e A322337   (16,8)   (12,8,4)   (12,6,4,2)
%e A322337   (18,6)   (12,9,3)
%e A322337   (20,4)   (14,6,4)
%e A322337   (21,3)   (14,8,2)
%e A322337   (22,2)   (15,6,3)
%e A322337   (14,10)  (16,6,2)
%e A322337            (18,4,2)
%e A322337            (12,10,2)
%t A322337 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322337 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 A322337 twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
%t A322337 Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,twoedQ[primeMS/@#]]&]],{n,30}]
%Y A322337 Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.
%K A322337 nonn
%O A322337 1,10
%A A322337 _Gus Wiseman_, Dec 04 2018