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

%I A370641 #12 Mar 28 2025 15:26:46
%S A370641 0,1,1,2,3,5,9,15,32,45,67,98,141,197,263,358,1201,1493,1920,2482,
%T A370641 3123,3967,4884,6137,7584,9369,11169,13664,15818,19152,22418,26905,
%U A370641 151286,173409,202171,237572,273651,320040,367792,428747,485697,562620,637043,734738,815492
%N A370641 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.
%C A370641 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 A370641 Also choices of A070939(n) elements of {1..n} containing n such that it is possible to choose a different binary index of each.
%e A370641 The a(0) = 0 through a(7) = 15 subsets:
%e A370641   .  {1}  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {1,2,7}
%e A370641                  {2,3}  {1,3,4}  {1,3,5}  {1,3,6}  {1,3,7}
%e A370641                         {2,3,4}  {2,3,5}  {1,4,6}  {1,4,7}
%e A370641                                  {2,4,5}  {1,5,6}  {1,5,7}
%e A370641                                  {3,4,5}  {2,3,6}  {1,6,7}
%e A370641                                           {2,5,6}  {2,3,7}
%e A370641                                           {3,4,6}  {2,4,7}
%e A370641                                           {3,5,6}  {2,5,7}
%e A370641                                           {4,5,6}  {2,6,7}
%e A370641                                                    {3,4,7}
%e A370641                                                    {3,5,7}
%e A370641                                                    {3,6,7}
%e A370641                                                    {4,5,7}
%e A370641                                                    {4,6,7}
%e A370641                                                    {5,6,7}
%t A370641 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A370641 Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}],MemberQ[#,n] && Length[Union[Sort/@Select[Tuples[bpe/@#], UnsameQ@@#&]]]>0&]],{n,0,25}]
%Y A370641 A version for set-systems is A368601.
%Y A370641 For prime indices we have A370590, without n A370585, see also A370591.
%Y A370641 This is the maximal case of A370636 requiring n, complement A370637.
%Y A370641 This is the maximal case of A370639, complement A370589.
%Y A370641 Without requiring n we have A370640.
%Y A370641 Dominated by A370819.
%Y A370641 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A370641 A058891 counts set-systems, A003465 covering, A323818 connected.
%Y A370641 A070939 gives length of binary expansion.
%Y A370641 A096111 gives product of binary indices.
%Y A370641 A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
%Y A370641 A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
%Y A370641 Cf. A133686, A326031, A326702, A357812, A367905, A368100, A368109, A370586, A370638, A370642.
%K A370641 nonn
%O A370641 0,4
%A A370641 _Gus Wiseman_, Mar 11 2024
%E A370641 More terms from _Jinyuan Wang_, Mar 28 2025