A329632 Number of connected integer partitions of n whose distinct parts are pairwise indivisible.

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 6, 4, 6, 1, 9, 2, 10, 6, 13, 3, 15, 6, 18, 8, 22, 9, 29, 10, 30, 20, 40, 22, 48, 24, 57, 36, 68

Offset: 0

Author

Gus Wiseman, Nov 18 2019

Keywords

• nonn
• more

Comments

Given an integer partition y of length k, let G(y) be the simple labeled graph with vertices {1..k} and edges between any two vertices i, j such that GCD(y_i, y_j) > 1. For example, G(6,14,15,35) is a 4-cycle. A partition y is said to be connected if G(y) is a connected graph.

Examples

			The a(n) partitions for n = 1, 4, 6, 10, 12, 14:
  (1)  (4)    (6)      (10)         (12)           (14)
       (2,2)  (3,3)    (5,5)        (6,6)          (7,7)
              (2,2,2)  (6,4)        (4,4,4)        (8,6)
                       (2,2,2,2,2)  (3,3,3,3)      (10,4)
                                    (2,2,2,2,2,2)  (6,4,4)
                                                   (2,2,2,2,2,2,2)
		

Crossrefs

The Heinz numbers of these partitions are given by A329559.

The strict version is A304717.

Connected partitions are A218970.

Pairwise indivisible partitions are A305148.

Cf. A048143, A286518, A286520, A304714, A304716, A305078.

Programs

Mathematica

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],stableQ[#,Divisible]&&Length[zsm[#]]<=1&]],{n,0,30}]