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A322337
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Number of strict 2-edge-connected integer partitions of n.
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8
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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, 32, 17, 39, 16, 59, 14, 58, 33, 75, 28, 103, 35, 106, 71, 125, 63, 174, 81, 192, 127, 220, 130, 294, 170, 325, 237, 378, 257, 504
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OFFSET
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1,10
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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.
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LINKS
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EXAMPLE
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The a(24) = 18 strict 2-edge-connected integer partitions of 24:
(15,9) (10,8,6) (10,8,4,2)
(16,8) (12,8,4) (12,6,4,2)
(18,6) (12,9,3)
(20,4) (14,6,4)
(21,3) (14,8,2)
(22,2) (15,6,3)
(14,10) (16,6,2)
(18,4,2)
(12,10,2)
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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], And[UnsameQ@@#, twoedQ[primeMS/@#]]&]], {n, 30}]
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CROSSREFS
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Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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