A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

1, 1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 4, 8, 4, 7, 7, 12, 10, 14, 12, 15, 19, 19, 21, 32, 27, 33, 40, 46, 47, 61, 52, 75, 89, 95, 104, 129, 129, 149, 176, 188, 208, 249, 257, 296, 341, 373, 394, 476, 496, 552

Offset: 0

Author

Gus Wiseman, Apr 02 2024

Keywords

• nonn
• more

Comments

These partitions are ranked by A371445.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

A binary carry of two positive integers is an overlap of binary indices. An integer partition is binary carry-connected iff the graph with one vertex for each part and edges corresponding to binary carries is connected.

A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

Examples

			The a(12) = 8 through a(14) = 7 partitions:
  (12)             (13)                         (14)
  (6,6)            (10,3)                       (7,7)
  (9,3)            (5,5,3)                      (9,5)
  (4,4,4)          (1,1,1,1,1,1,1,1,1,1,1,1,1)  (6,5,3)
  (6,3,3)                                       (5,3,3,3)
  (3,3,3,3)                                     (2,2,2,2,2,2,2)
  (2,2,2,2,2,2)                                 (1,1,1,1,1,1,1,1,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)
		

Crossrefs

The first condition (carry-connected) is A325098.

The second condition (stable) is A325109.

Ranks for binary indices of binary indices are A326750 = A326704 /\ A326749.

Ranks for prime indices of prime indices are A329559 = A305078 /\ A316476.

Ranks for prime indices of binary indices are A371294 = A087086 /\ A371291.

Ranks for binary indices of prime indices are A371445 = A325118 /\ A371455.

A001187 counts connected graphs.

A007718 counts non-isomorphic connected multiset partitions.

A048143 counts connected antichains of sets.

A048793 lists binary indices, reverse A272020, length A000120, sum A029931.

A070939 gives length of binary expansion.

A326964 counts connected set-systems, covering A323818.

Cf. A019565, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

Programs

Mathematica

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], stableQ[bix/@Union[#],SubsetQ]&&Length[csm[bix/@#]]<=1&]],{n,0,30}]