A370640 - OEIS

%I #6 Mar 10 2024 21:23:26

%S 1,1,1,3,3,8,17,32,32,77,144,242,383,580,843,1201,1201,2694,4614,7096,

%T 10219,14186,19070,25207,32791,42160

%N Number of maximal subsets of {1..n} such that it is possible to choose 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.

%C Also choices of A029837(n) elements of {1..n} such that it is possible to choose a different binary index of each.

%e The a(0) = 1 through a(6) = 17 subsets:

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

%e                   {1,3}  {1,3,4}  {1,2,5}  {1,2,5}

%e                   {2,3}  {2,3,4}  {1,3,4}  {1,2,6}

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

%e                                   {2,3,4}  {1,3,5}

%e                                   {2,3,5}  {1,3,6}

%e                                   {2,4,5}  {1,4,6}

%e                                   {3,4,5}  {1,5,6}

%e                                            {2,3,4}

%e                                            {2,3,5}

%e                                            {2,3,6}

%e                                            {2,4,5}

%e                                            {2,5,6}

%e                                            {3,4,5}

%e                                            {3,4,6}

%e                                            {3,5,6}

%e                                            {4,5,6}

%e The a(0) = 1 through a(6) = 17 set-systems:

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

%e                  {1}{12}  {1}{12}{3}  {1}{12}{3}   {1}{12}{3}

%e                  {2}{12}  {2}{12}{3}  {1}{2}{13}   {1}{2}{13}

%e                                       {2}{12}{3}   {1}{2}{23}

%e                                       {2}{3}{13}   {1}{3}{23}

%e                                       {1}{12}{13}  {2}{12}{3}

%e                                       {12}{3}{13}  {2}{3}{13}

%e                                       {2}{12}{13}  {1}{12}{13}

%e                                                    {1}{12}{23}

%e                                                    {1}{13}{23}

%e                                                    {12}{3}{13}

%e                                                    {12}{3}{23}

%e                                                    {2}{12}{13}

%e                                                    {2}{12}{23}

%e                                                    {2}{13}{23}

%e                                                    {3}{13}{23}

%e                                                    {12}{13}{23}

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

%t Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]

%Y Dominated by A357812.

%Y The version for set-systems is A368601, max of A367902 (complement A367903).

%Y For prime indices we have A370585, with n A370590, see also A370591.

%Y This is the maximal case of A370636 (complement A370637).

%Y The case of a unique choice is A370638.

%Y The case containing n is A370641, non-maximal A370639.

%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 A307984 counts Q-bases of logarithms of positive integers.

%Y A355741 counts choices of a prime factor of each prime index.

%Y Cf. A133686, A326031, A326702, A367905, A367909, A367912, A368109, A368110, A370592, A370642.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Mar 10 2024