A322387 Number of 2-vertex-connected integer partitions of n.

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).

Links

Table of n, a(n) for n=1..42.

Wikipedia, k-vertex-connected graph

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}]

Crossrefs

Cf. A013922, A095983, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322388, A322389, A322390.

Sequence in context: A274317 A105358 A240691 * A336173 A072361 A341219

Adjacent sequences: A322384 A322385 A322386 * A322388 A322389 A322390

Keyword

nonn,more

Author

Gus Wiseman, Dec 05 2018

Extensions

a(41)-a(42) from Jinyuan Wang, Jun 20 2020

Status

approved