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A322391
Number of integer partitions of n with edge-connectivity 1.
5
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 9, 3, 14, 8, 17, 13, 35, 17, 49, 35, 67, 53, 114, 69
OFFSET
1,11
COMMENTS
The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.
EXAMPLE
The a(20) = 8 integer partitions:
(20),
(12,3,3,2), (9,6,3,2), (8,6,3,3),
(6,4,4,3,3),
(6,4,3,3,2,2), (6,3,3,3,3,2),
(6,3,3,2,2,2,2).
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1, 0, Length[y]-Max@@Length/@Select[Union[Subsets[y]], Length[csm[primeMS/@#]]!=1&]];
Table[Length[Select[IntegerPartitions[n], edgeConn[#]==1&]], {n, 20}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 05 2018
STATUS
approved