A322337 Number of strict 2-edge-connected integer partitions of n.

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

Author

Gus Wiseman, Dec 04 2018

Keywords

• nonn

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.

Examples

			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)
		

Links

• Wikipedia, k-edge-connected graph

Crossrefs

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.

Programs

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}]