A370636 - OEIS

%I #12 Mar 09 2024 16:40:38

%S 1,2,4,7,14,24,39,61,122,203,315,469,676,952,1307,1771,3542,5708,8432,

%T 11877,16123,21415,27835,35757,45343,57010,70778,87384,106479,129304,

%U 155802,187223,374446,588130,835800,1124981,1456282,1841361,2281772,2791896,3367162

%N Number of 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.

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

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%F a(2^n - 1) = A367902(n).

%F Partial sums of A370639.

%e The a(0) = 1 through a(4) = 14 subsets:

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

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

%e            {2}    {2}    {2}

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

%e                   {1,2}  {4}

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

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

%e                          {1,4}

%e                          {2,3}

%e                          {2,4}

%e                          {3,4}

%e                          {1,2,4}

%e                          {1,3,4}

%e                          {2,3,4}

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

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

%Y Simple graphs of this type are counted by A133686, covering A367869.

%Y Unlabeled graphs of this type are counted by A134964, complement A140637.

%Y Simple graphs not of this type are counted by A367867, covering A367868.

%Y Set systems of this type are counted by A367902, ranks A367906.

%Y Set systems not of this type are counted by A367903, ranks A367907.

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

%Y Unlabeled multiset partitions of this type are A368098, complement A368097.

%Y A version for MM-numbers of multisets is A368100, complement A355529.

%Y Factorizations are counted by A368414/A370814, complement A368413/A370813.

%Y For prime indices we have A370582, differences A370586.

%Y The complement for prime indices is A370583, differences A370587.

%Y The complement is A370637, differences A370589, without ones A370643.

%Y The case of a unique choice is A370638, maxima A370640, differences A370641.

%Y First differences are A370639.

%Y The minimal case of the complement is A370642, without ones A370644.

%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 A326031 gives weight of the set-system with BII-number n.

%Y Cf. A000612, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368095, A368109.

%K nonn

%O 0,2

%A _Gus Wiseman_, Mar 08 2024

%E a(19)-a(40) from _Alois P. Heinz_, Mar 09 2024