OFFSET
0,3
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.
EXAMPLE
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)
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}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Apr 02 2024
STATUS
approved