OFFSET
0,2
COMMENTS
A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. A subset is binary carry-connected if the graph whose vertices are the elements and whose edges are binary carries is connected.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1023
FORMULA
EXAMPLE
The a(0) = 1 through a(4) = 8 subsets:
{} {} {} {} {}
{1} {1} {1} {1}
{2} {2} {2}
{3} {3}
{1,3} {4}
{2,3} {1,3}
{1,2,3} {2,3}
{1,2,3}
MAPLE
h:= proc(n, s) local i, m; m:= n;
for i in s do m:= Bits[Or](m, i) od; {m}
end:
g:= (n, s)-> (w-> `if`(w={}, s union {n}, s minus w union
h(n, w)))(select(x-> Bits[And](n, x)>0, s)):
b:= proc(n, s) option remember; `if`(n=0,
`if`(nops(s)>1, 0, 1), b(n-1, s)+b(n-1, g(n, s)))
end:
a:= n-> b(n, {}):
seq(a(n), n=0..35); # Alois P. Heinz, Mar 31 2019
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[Subsets[Range[n]], Length[csm[binpos/@#]]<=1&]], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 28 2019
EXTENSIONS
a(16)-a(33) from Alois P. Heinz, Mar 31 2019
STATUS
approved