A370638 - OEIS

%I #9 Mar 10 2024 15:12:07

%S 1,2,4,6,12,19,30,45,90,147,230,343,504,716,994,1352,2704,4349,6469

%N Number of subsets of {1..n} such that a unique set can be obtained by choosing 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.

%F a(2^n - 1) = A370818(n).

%e 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 The a(0) = 1 through a(5) = 19 subsets:

%e   {}  {}   {}     {}     {}       {}

%e       {1}  {1}    {1}    {1}      {1}

%e            {2}    {2}    {2}      {2}

%e            {1,2}  {1,2}  {4}      {4}

%e                   {1,3}  {1,2}    {1,2}

%e                   {2,3}  {1,3}    {1,3}

%e                          {1,4}    {1,4}

%e                          {2,3}    {1,5}

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

%e                          {1,2,4}  {2,4}

%e                          {1,3,4}  {4,5}

%e                          {2,3,4}  {1,2,4}

%e                                   {1,2,5}

%e                                   {1,3,4}

%e                                   {1,3,5}

%e                                   {2,3,4}

%e                                   {2,3,5}

%e                                   {2,4,5}

%e                                   {3,4,5}

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

%t Table[Length[Select[Subsets[Range[n]],Length[Union[Sort /@ Select[Tuples[bpe/@#],UnsameQ@@#&]]]==1&]],{n,0,10}]

%Y Set systems of this type are counted by A367904, ranks A367908.

%Y A version for MM-numbers of multisets is A368101.

%Y For prime indices we have A370584.

%Y This is the unique version of A370636, complement A370637.

%Y The maximal case is A370640, differences A370641.

%Y Factorizations of this type are counted by A370645.

%Y The case A370818 is the restriction to A000225.

%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 Cf. A133686, A134964, A326031, A326702, A367772, A367867, A367905, A367909, A367912, A368109.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Mar 09 2024