login
Number of maximal subsets of {1...n} with no binary carries.
17

%I #12 Jul 27 2019 14:57:51

%S 1,1,1,2,2,3,4,5,5,7,9,10,12,13,14,15,15,20,25,27,32,34,36,37,42,44,

%T 46,47,49,50,51,52,52,67,82,87,102,107,112,114,129,134,139,141,146,

%U 148,150,151,166,171,176,178,183,185,187,188,193,195,197,198,200,201

%N Number of maximal subsets of {1...n} with no binary carries.

%C A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

%H Alois P. Heinz, <a href="/A325096/b325096.txt">Table of n, a(n) for n = 0..8192</a>

%F a(2^n - 1) = A000110(n).

%e The a(1) = 1 through a(9) = 7 maximal subsets:

%e {1} {12} {3} {34} {25} {16} {7} {78} {69}

%e {12} {124} {34} {25} {16} {168} {78}

%e {124} {34} {25} {258} {168}

%e {124} {34} {348} {249}

%e {124} {1248} {258}

%e {348}

%e {1248}

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

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];

%t Table[Length[maxim[Select[Subsets[Range[n]],stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]]],{n,0,10}]

%Y Cf. A000110, A019565, A050315, A080572, A247935, A267610, A267700.

%Y Cf. A325095, A325099, A325100, A325101, A325103, A325104, A325107.

%K nonn,look

%O 0,4

%A _Gus Wiseman_, Mar 27 2019

%E a(15)-a(61) from _Alois P. Heinz_, Mar 28 2019