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A322387 Number of 2-vertex-connected integer partitions of n.

%I A322387 #10 Jun 20 2020 09:00:39
%S A322387 0,0,0,0,0,1,0,1,1,3,1,6,2,10,8,13,9,26,14,35,28,50,37,77,54,101,84,
%T A322387 138,110,205,149,252,222,335,287,455,375,577,522,740,657,985
%N A322387 Number of 2-vertex-connected integer partitions of n.
%C A322387 An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).
%H A322387 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-vertex-connected_graph">k-vertex-connected graph</a>
%e A322387 The a(14) = 10 2-vertex-connected integer partitions:
%e A322387   (14)  (8,6)   (6,4,4)   (6,3,3,2)  (6,2,2,2,2)
%e A322387         (10,4)  (6,6,2)   (6,4,2,2)
%e A322387         (12,2)  (10,2,2)
%t A322387 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322387 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 A322387 vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
%t A322387 Table[Length[Select[IntegerPartitions[n],vertConn[#]>1&]],{n,30}]
%Y A322387 Cf. A013922, A095983, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322388, A322389, A322390.
%K A322387 nonn,more
%O A322387 1,10
%A A322387 _Gus Wiseman_, Dec 05 2018
%E A322387 a(41)-a(42) from _Jinyuan Wang_, Jun 20 2020