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A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

%I A370639 #9 Mar 09 2024 15:28:28
%S A370639 0,1,2,3,7,10,15,22,61,81,112,154,207,276,355,464,1771,2166,2724,3445,
%T A370639 4246,5292,6420,7922,9586,11667,13768,16606,19095,22825,26498,31421,
%U A370639 187223,213684,247670,289181,331301,385079,440411,510124,575266,662625,747521
%N A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.
%C A370639 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.
%H A370639 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%F A370639 First differences of A370636.
%e A370639 The a(0) = 0 through a(6) = 15 subsets:
%e A370639   .  {1}  {2}    {3}    {4}      {5}      {6}
%e A370639           {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
%e A370639                  {2,3}  {2,4}    {2,5}    {2,6}
%e A370639                         {3,4}    {3,5}    {3,6}
%e A370639                         {1,2,4}  {4,5}    {4,6}
%e A370639                         {1,3,4}  {1,2,5}  {5,6}
%e A370639                         {2,3,4}  {1,3,5}  {1,2,6}
%e A370639                                  {2,3,5}  {1,3,6}
%e A370639                                  {2,4,5}  {1,4,6}
%e A370639                                  {3,4,5}  {1,5,6}
%e A370639                                           {2,3,6}
%e A370639                                           {2,5,6}
%e A370639                                           {3,4,6}
%e A370639                                           {3,5,6}
%e A370639                                           {4,5,6}
%t A370639 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A370639 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]
%Y A370639 Simple graphs of this type are counted by A133686, covering A367869.
%Y A370639 Unlabeled graphs of this type are counted by A134964, complement A140637.
%Y A370639 Simple graphs not of this type are counted by A367867, covering A367868.
%Y A370639 Set systems of this type are counted by A367902, ranks A367906.
%Y A370639 Set systems not of this type are counted by A367903, ranks A367907.
%Y A370639 Set systems uniquely of this type are counted by A367904, ranks A367908.
%Y A370639 Unlabeled multiset partitions of this type are A368098, complement A368097.
%Y A370639 A version for MM-numbers of multisets is A368100, complement A355529.
%Y A370639 Factorizations of this type are A368414/A370814, complement A368413/A370813.
%Y A370639 For prime instead of binary indices we have A370586, differences of A370582.
%Y A370639 The complement for prime indices is A370587, differences of A370583.
%Y A370639 The complement is counted by A370589, differences of A370637.
%Y A370639 Partial sums are A370636.
%Y A370639 The complement has partial sums A370637/A370643, minima A370642/A370644.
%Y A370639 The case of a unique choice is A370641, differences of A370638.
%Y A370639 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A370639 A058891 counts set-systems, A003465 covering, A323818 connected.
%Y A370639 A070939 gives length of binary expansion.
%Y A370639 A096111 gives product of binary indices.
%Y A370639 A326031 gives weight of the set-system with BII-number n.
%Y A370639 Cf. A000612, A326702, A355739, A355740, A367770, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.
%K A370639 nonn
%O A370639 0,3
%A A370639 _Gus Wiseman_, Mar 08 2024
%E A370639 a(19)-a(42) from _Alois P. Heinz_, Mar 09 2024