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A322337
Number of strict 2-edge-connected integer partitions of n.
8
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
OFFSET
1,10
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 04 2018
STATUS
approved