A325099 Number of binary carry-connected strict integer partitions of n.

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, 125, 131, 165, 184, 237, 251, 291, 315, 383, 408, 486, 536, 663, 714, 832, 912, 1104, 1195, 1405, 1554, 1877, 2046, 2348, 2559, 2998, 3256, 3730, 4084, 4793, 5230, 5938

Offset: 0

Author

Gus Wiseman, Mar 28 2019

Keywords

• nonn

Comments

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.

Examples

			The a(1) = 1 through a(11) = 6 strict partitions (A = 10, B = 11):
  (1)  (2)  (3)  (4)   (5)   (6)    (7)  (8)   (9)    (A)    (B)
                 (31)  (32)  (51)        (53)  (54)   (64)   (65)
                             (321)       (62)  (63)   (73)   (74)
                                         (71)  (72)   (91)   (632)
                                               (531)  (532)  (731)
                                                      (541)  (5321)
                                                      (631)
                                                      (721)
		

Crossrefs

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

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

Programs

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[csm[binpos/@#]]<=1&]],{n,0,30}]