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A370638 Number of subsets of {1..n} such that a unique set can be obtained by choosing a different binary index of each element.

%I A370638 #13 Mar 28 2025 14:12:56
%S A370638 1,2,4,6,12,19,30,45,90,147,230,343,504,716,994,1352,2704,4349,6469,
%T A370638 9162,12585,16862,22122,28617,36653,46431,58075,72097,88456,107966,
%U A370638 130742,157647,315294,494967,704753,950080,1234301,1565165,1945681,2387060,2890368,3470798
%N A370638 Number of subsets of {1..n} such that a unique set can be obtained by choosing a different binary index of each element.
%C A370638 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 A370638 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%F A370638 a(2^n - 1) = A370818(n).
%e A370638 The set {3,4} has binary indices {{1,2},{3}}, with two choices {1,3}, {2,3}, so is not counted under a(4).
%e A370638 The a(0) = 1 through a(5) = 19 subsets:
%e A370638   {}  {}   {}     {}     {}       {}
%e A370638       {1}  {1}    {1}    {1}      {1}
%e A370638            {2}    {2}    {2}      {2}
%e A370638            {1,2}  {1,2}  {4}      {4}
%e A370638                   {1,3}  {1,2}    {1,2}
%e A370638                   {2,3}  {1,3}    {1,3}
%e A370638                          {1,4}    {1,4}
%e A370638                          {2,3}    {1,5}
%e A370638                          {2,4}    {2,3}
%e A370638                          {1,2,4}  {2,4}
%e A370638                          {1,3,4}  {4,5}
%e A370638                          {2,3,4}  {1,2,4}
%e A370638                                   {1,2,5}
%e A370638                                   {1,3,4}
%e A370638                                   {1,3,5}
%e A370638                                   {2,3,4}
%e A370638                                   {2,3,5}
%e A370638                                   {2,4,5}
%e A370638                                   {3,4,5}
%t A370638 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A370638 Table[Length[Select[Subsets[Range[n]],Length[Union[Sort /@ Select[Tuples[bpe/@#],UnsameQ@@#&]]]==1&]],{n,0,10}]
%Y A370638 Set systems of this type are counted by A367904, ranks A367908.
%Y A370638 A version for MM-numbers of multisets is A368101.
%Y A370638 For prime indices we have A370584.
%Y A370638 This is the unique version of A370636, complement A370637.
%Y A370638 The maximal case is A370640, differences A370641.
%Y A370638 Factorizations of this type are counted by A370645.
%Y A370638 The case A370818 is the restriction to A000225.
%Y A370638 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A370638 A058891 counts set-systems, A003465 covering, A323818 connected.
%Y A370638 A070939 gives length of binary expansion.
%Y A370638 A096111 gives product of binary indices.
%Y A370638 Cf. A133686, A134964, A326031, A326702, A367772, A367867, A367905, A367909, A367912, A368109.
%K A370638 nonn
%O A370638 0,2
%A A370638 _Gus Wiseman_, Mar 09 2024
%E A370638 More terms from _Jinyuan Wang_, Mar 28 2025