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A325099
Number of binary carry-connected strict integer partitions of n.
6
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
0,5
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.
EXAMPLE
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)
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 28 2019
STATUS
approved