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%I #4 Dec 06 2018 06:52:24
%S 1,1,1,1,1,1,1,1,1,1,2,1,3,2,3,3,9,3,14,8,17,13,35,17,49,35,67,53,114,
%T 69
%N Number of integer partitions of n with edge-connectivity 1.
%C The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>
%e The a(20) = 8 integer partitions:
%e (20),
%e (12,3,3,2), (9,6,3,2), (8,6,3,3),
%e (6,4,4,3,3),
%e (6,4,3,3,2,2), (6,3,3,3,3,2),
%e (6,3,3,2,2,2,2).
%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 edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]];
%t Table[Length[Select[IntegerPartitions[n],edgeConn[#]==1&]],{n,20}]
%Y Cf. A007718, A013922, A054921, A095983, A218970, A275307, A304716, A305078, A305079, A322335, A322336, A322387, A322390.
%K nonn,more
%O 1,11
%A _Gus Wiseman_, Dec 05 2018
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