%I #6 Jul 27 2019 14:57:51

%S 1,1,1,1,2,2,3,1,4,5,8,6,11,11,15,13,18,20,30,29,43,49,68,66,84,94,

%T 125,131,165,184,237,251,291,315,383,408,486,536,663,714,832,912,1104,

%U 1195,1405,1554,1877,2046,2348,2559,2998,3256,3730,4084,4793,5230,5938

%N Number of binary carry-connected strict integer partitions of n.

%C A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

%e The a(1) = 1 through a(11) = 6 strict partitions (A = 10, B = 11):

%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)

%e (31) (32) (51) (53) (54) (64) (65)

%e (321) (62) (63) (73) (74)

%e (71) (72) (91) (632)

%e (531) (532) (731)

%e (541) (5321)

%e (631)

%e (721)

%t binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t 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]]]]]]]]];

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[csm[binpos/@#]]<=1&]],{n,0,30}]

%Y Cf. A050315, A080572, A247935, A267610, A267700.

%Y Cf. A325095, A325096, A325098, A325104, A325106, A325110, A325119.

%K nonn

%O 0,5

%A _Gus Wiseman_, Mar 28 2019