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A322335
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Number of 2-edge-connected integer partitions of n.
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14
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0, 0, 0, 1, 0, 3, 0, 4, 2, 7, 0, 13, 0, 15, 8, 21, 1, 37, 2, 45, 18, 58, 8, 95, 19, 109, 45, 150, 38, 232, 59, 268, 129, 357, 155, 523, 203, 633, 359, 852, 431, 1185, 609, 1464, 969
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OFFSET
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1,6
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COMMENTS
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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. For example (6,6,3,2) is 2-edge-connected but (6,3,2) is not.
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LINKS
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EXAMPLE
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The a(14) = 15 2-edge-connected integer partitions of 14:
(7,7) (6,4,4) (4,4,4,2) (4,4,2,2,2) (4,2,2,2,2,2) (2,2,2,2,2,2,2)
(8,6) (6,6,2) (6,4,2,2) (6,2,2,2,2)
(10,4) (8,4,2) (8,2,2,2)
(12,2) (10,2,2)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
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]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1, And@@Table[Length[csm[Delete[sys, i]]]==1, {i, Length[sys]}]];
Table[Length[Select[IntegerPartitions[n], twoedQ[primeMS/@#]&]], {n, 30}]
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CROSSREFS
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Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A322336, A322337, A322338.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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