A325107 Number of subsets of {1...n} with no binary containments.

1, 2, 4, 5, 10, 13, 18, 19, 38, 52, 77, 83, 133, 147, 166, 167, 334, 482, 764, 848, 1465, 1680, 1987, 2007, 3699, 4413, 5488, 5572, 7264, 7412, 7579, 7580, 15160, 22573, 37251, 42824, 77387, 92863, 116453, 118461, 227502, 286775, 382573, 392246, 555661, 574113

Offset

0,2

Comments

A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..129, (terms up to a(71) from Alois P. Heinz)

Formula

a(2^n - 1) = A014466(n).

Example

The a(0) = 1 through a(6) = 18 subsets:
  {}  {}   {}     {}     {}       {}       {}
      {1}  {1}    {1}    {1}      {1}      {1}
           {2}    {2}    {2}      {2}      {2}
           {1,2}  {3}    {3}      {3}      {3}
                  {1,2}  {4}      {4}      {4}
                         {1,2}    {5}      {5}
                         {1,4}    {1,2}    {6}
                         {2,4}    {1,4}    {1,2}
                         {3,4}    {2,4}    {1,4}
                         {1,2,4}  {2,5}    {1,6}
                                  {3,4}    {2,4}
                                  {3,5}    {2,5}
                                  {1,2,4}  {3,4}
                                           {3,5}
                                           {3,6}
                                           {5,6}
                                           {1,2,4}
                                           {3,5,6}

Maple

c:= proc() option remember; local i, x, y;
      x, y:= map(n-> Bits[Split](n), [args])[];
      for i to nops(x) do
        if x[i]=1 and y[i]=0 then return false fi
      od; true
    end:
b:= proc(n, s) option remember; `if`(n=0, 1, b(n-1, s)+
     `if`(ormap(i-> c(n, i), s), 0, b(n-1, s union {n})))
    end:
a:= n-> b(n, {}):
seq(a(n), n=0..34);  # Alois P. Heinz, Mar 28 2019

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[Subsets[Range[n]], stableQ[#, SubsetQ[binpos[#1], binpos[#2]]&]&]], {n, 0, 13}]

Crossrefs

Cf. A006126, A014466, A019565, A267610.

Cf. A325095, A325096, A325101, A325105, A325106, A325108, A325109, A325110.

Sequence in context: A283204 A105138 A326311 * A064383 A018360 A133585

Adjacent sequences: A325104 A325105 A325106 * A325108 A325109 A325110

Keyword

nonn

Author

Gus Wiseman, Mar 28 2019

Extensions

a(16)-a(45) from Alois P. Heinz, Mar 28 2019

Status

approved