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A322387
Number of 2-vertex-connected integer partitions of n.
13
0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 1, 6, 2, 10, 8, 13, 9, 26, 14, 35, 28, 50, 37, 77, 54, 101, 84, 138, 110, 205, 149, 252, 222, 335, 287, 455, 375, 577, 522, 740, 657, 985
OFFSET
1,10
COMMENTS
An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).
EXAMPLE
The a(14) = 10 2-vertex-connected integer partitions:
(14) (8,6) (6,4,4) (6,3,3,2) (6,2,2,2,2)
(10,4) (6,6,2) (6,4,2,2)
(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]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1, 0, Min@@Length/@Select[Subsets[Union@@primeMS/@y], Function[del, Length[csm[DeleteCases[DeleteCases[primeMS/@y, Alternatives@@del, {2}], {}]]]!=1]]];
Table[Length[Select[IntegerPartitions[n], vertConn[#]>1&]], {n, 30}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 05 2018
EXTENSIONS
a(41)-a(42) from Jinyuan Wang, Jun 20 2020
STATUS
approved