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A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.

%I A370589 #8 Mar 09 2024 20:40:36
%S A370589 0,0,0,1,1,6,17,42,67,175,400,870,1841,3820,7837,15920,30997,63370,
%T A370589 128348,258699,520042,1043284,2090732,4186382,8379022,16765549,
%U A370589 33540664,67092258,134198633,268412631,536844414,1073710403,2147296425,4294753612,8589686922,17179580003
%N A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.
%C A370589 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 A370589 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A370589 The binary indices of {1,4,5} are {{1},{3},{1,3}}, from which it is not possible to choose three different elements, so S is counted under a(3).
%e A370589 The binary indices of S = {1,6,8,9} are {{1},{2,3},{4},{1,4}}, from which it is not possible to choose four different elements, so S is counted under a(9).
%e A370589 The a(0) = 0 through a(6) = 17 subsets:
%e A370589   .  .  .  {1,2,3}  {1,2,3,4}  {1,4,5}      {2,4,6}
%e A370589                                {1,2,3,5}    {1,2,3,6}
%e A370589                                {1,2,4,5}    {1,2,4,6}
%e A370589                                {1,3,4,5}    {1,2,5,6}
%e A370589                                {2,3,4,5}    {1,3,4,6}
%e A370589                                {1,2,3,4,5}  {1,3,5,6}
%e A370589                                             {1,4,5,6}
%e A370589                                             {2,3,4,6}
%e A370589                                             {2,3,5,6}
%e A370589                                             {2,4,5,6}
%e A370589                                             {3,4,5,6}
%e A370589                                             {1,2,3,4,6}
%e A370589                                             {1,2,3,5,6}
%e A370589                                             {1,2,4,5,6}
%e A370589                                             {1,3,4,5,6}
%e A370589                                             {2,3,4,5,6}
%e A370589                                             {1,2,3,4,5,6}
%t A370589 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A370589 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]
%Y A370589 Simple graphs not of this type are counted by A133686, covering A367869.
%Y A370589 Unlabeled graphs of this type are counted by A140637, complement A134964.
%Y A370589 Simple graphs of this type are counted by A367867, covering A367868.
%Y A370589 Set systems not of this type are counted by A367902, ranks A367906.
%Y A370589 Set systems of this type are counted by A367903, ranks A367907.
%Y A370589 Set systems uniquely not of this type are counted by A367904, ranks A367908.
%Y A370589 Unlabeled multiset partitions of this type are A368097, complement A368098.
%Y A370589 A version for MM-numbers of multisets is A355529, complement A368100.
%Y A370589 Factorizations are counted by A368413/A370813, complement A368414/A370814.
%Y A370589 The complement for prime indices is A370586, differences of A370582.
%Y A370589 For prime indices we have A370587, differences of A370583.
%Y A370589 Partial sums are A370637/A370643, minima A370642/A370644.
%Y A370589 The complement is counted by A370639, partial sums A370636.
%Y A370589 The version for a unique choice is A370641, partial sums A370638.
%Y A370589 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A370589 A058891 counts set-systems, A003465 covering, A323818 connected.
%Y A370589 A070939 gives length of binary expansion.
%Y A370589 A096111 gives product of binary indices.
%Y A370589 A326031 gives weight of the set-system with BII-number n.
%Y A370589 Cf. A000612, A072639, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.
%K A370589 nonn
%O A370589 0,6
%A A370589 _Gus Wiseman_, Mar 08 2024
%E A370589 a(19)-a(35) from _Alois P. Heinz_, Mar 09 2024