A370641 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

0, 1, 1, 2, 3, 5, 9, 15, 32, 45, 67, 98, 141, 197, 263, 358, 1201, 1493, 1920, 2482, 3123, 3967, 4884, 6137, 7584, 9369

Offset

0,4

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.

Also choices of A029837(n) elements of {1..n} containing n such that it is possible to choose a different binary index of each.

Links

Table of n, a(n) for n=0..25.

Example

The a(0) = 0 through a(7) = 15 subsets:
  .  {1}  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {1,2,7}
                 {2,3}  {1,3,4}  {1,3,5}  {1,3,6}  {1,3,7}
                        {2,3,4}  {2,3,5}  {1,4,6}  {1,4,7}
                                 {2,4,5}  {1,5,6}  {1,5,7}
                                 {3,4,5}  {2,3,6}  {1,6,7}
                                          {2,5,6}  {2,3,7}
                                          {3,4,6}  {2,4,7}
                                          {3,5,6}  {2,5,7}
                                          {4,5,6}  {2,6,7}
                                                   {3,4,7}
                                                   {3,5,7}
                                                   {3,6,7}
                                                   {4,5,7}
                                                   {4,6,7}
                                                   {5,6,7}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[n], {IntegerLength[n, 2]}], MemberQ[#, n] && Length[Union[Sort/@Select[Tuples[bpe/@#], UnsameQ@@#&]]]>0&]], {n, 0, 25}]

Crossrefs

A version for set-systems is A368601.

For prime indices we have A370590, without n A370585, see also A370591.

This is the maximal case of A370636 requiring n, complement A370637.

This is the maximal case of A370639, complement A370589.

Without requiring n we have A370640.

Dominated by A370819.

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.

A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.

A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.

Cf. A133686, A326031, A326702, A357812, A367905, A368100, A368109, A370586, A370638, A370642.

Sequence in context: A191701 A066726 A124642 * A269153 A232866 A011826

Adjacent sequences: A370638 A370639 A370640 * A370642 A370643 A370644

Keyword

nonn,more

Author

Gus Wiseman, Mar 11 2024

Status

approved