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A322335
Number of 2-edge-connected integer partitions of n.
14
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
OFFSET
1,6
COMMENTS
First differs from A108572 at a(17) = 1, A108572(17) = 0.
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.
EXAMPLE
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)
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], twoedQ[primeMS/@#]&]], {n, 30}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 04 2018
EXTENSIONS
a(42)-a(45) from Jinyuan Wang, Jun 20 2020
STATUS
approved