A370641 - OEIS

%I #7 Mar 11 2024 18:06:09

%S 0,1,1,2,3,5,9,15,32,45,67,98,141,197,263,358,1201,1493,1920,2482,

%T 3123,3967,4884,6137,7584,9369

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

%C 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.

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

%e The a(0) = 0 through a(7) = 15 subsets:

%e   .  {1}  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {1,2,7}

%e                  {2,3}  {1,3,4}  {1,3,5}  {1,3,6}  {1,3,7}

%e                         {2,3,4}  {2,3,5}  {1,4,6}  {1,4,7}

%e                                  {2,4,5}  {1,5,6}  {1,5,7}

%e                                  {3,4,5}  {2,3,6}  {1,6,7}

%e                                           {2,5,6}  {2,3,7}

%e                                           {3,4,6}  {2,4,7}

%e                                           {3,5,6}  {2,5,7}

%e                                           {4,5,6}  {2,6,7}

%e                                                    {3,4,7}

%e                                                    {3,5,7}

%e                                                    {3,6,7}

%e                                                    {4,5,7}

%e                                                    {4,6,7}

%e                                                    {5,6,7}

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

%t Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}],MemberQ[#,n] && Length[Union[Sort/@Select[Tuples[bpe/@#], UnsameQ@@#&]]]>0&]],{n,0,25}]

%Y A version for set-systems is A368601.

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

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

%Y This is the maximal case of A370639, complement A370589.

%Y Without requiring n we have A370640.

%Y Dominated by A370819.

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

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

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

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

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

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

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Mar 11 2024