A370638 Number of subsets of {1..n} such that a unique set can be obtained by choosing a different binary index of each element.

1, 2, 4, 6, 12, 19, 30, 45, 90, 147, 230, 343, 504, 716, 994, 1352, 2704, 4349, 6469

Offset

0,2

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Links

Table of n, a(n) for n=0..18.

Wikipedia, Axiom of choice.

Formula

a(2^n - 1) = A370818(n).

Example

The set {3,4} has binary indices {{1,2},{3}}, with two choices {1,3}, {2,3}, so is not counted under a(4).
The a(0) = 1 through a(5) = 19 subsets:
  {}  {}   {}     {}     {}       {}
      {1}  {1}    {1}    {1}      {1}
           {2}    {2}    {2}      {2}
           {1,2}  {1,2}  {4}      {4}
                  {1,3}  {1,2}    {1,2}
                  {2,3}  {1,3}    {1,3}
                         {1,4}    {1,4}
                         {2,3}    {1,5}
                         {2,4}    {2,3}
                         {1,2,4}  {2,4}
                         {1,3,4}  {4,5}
                         {2,3,4}  {1,2,4}
                                  {1,2,5}
                                  {1,3,4}
                                  {1,3,5}
                                  {2,3,4}
                                  {2,3,5}
                                  {2,4,5}
                                  {3,4,5}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[n]], Length[Union[Sort /@ Select[Tuples[bpe/@#], UnsameQ@@#&]]]==1&]], {n, 0, 10}]

Crossrefs

Set systems of this type are counted by A367904, ranks A367908.

A version for MM-numbers of multisets is A368101.

For prime indices we have A370584.

This is the unique version of A370636, complement A370637.

The maximal case is A370640, differences A370641.

Factorizations of this type are counted by A370645.

The case A370818 is the restriction to A000225.

A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

A058891 counts set-systems, A003465 covering, A323818 connected.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A133686, A134964, A326031, A326702, A367772, A367867, A367905, A367909, A367912, A368109.

Sequence in context: A357546 A294918 A347474 * A331872 A307067 A358101

Adjacent sequences: A370635 A370636 A370637 * A370639 A370640 A370641

Keyword

nonn,more

Author

Gus Wiseman, Mar 09 2024

Status

approved