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Number of sets that can be obtained by choosing a different binary index of each binary index of n.
8

%I #7 Dec 18 2023 08:28:42

%S 1,1,1,1,2,1,1,0,1,1,1,1,2,1,1,0,2,1,2,1,3,1,1,0,1,0,1,0,1,0,0,0,2,2,

%T 1,1,3,1,1,0,1,1,0,0,1,0,0,0,3,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,3,2,2,1,

%U 3,1,1,0,2,1,1,0,1,0,0,0,3,1,1,0,1,0,0

%N Number of sets that can be obtained by choosing a different binary index of each binary index of n.

%C A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

%e The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,3,2), (2,1,3), both permutations of {1,2,3}, so a(52) = 1.

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

%t Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]]],{n,0,100}]

%Y For sequences we have A367905, firsts A367910, sorted A367911.

%Y Positions of zeros are A367907.

%Y Without distinctness we have A367912, firsts A367913, sorted A367915.

%Y Positions of positive terms are A367906.

%Y For sequences without distinctness: A368109, firsts A368111, sorted A368112.

%Y Positions of first appearances are A368184, sorted A368185.

%Y A048793 lists binary indices, length A000120, sum A029931.

%Y A058891 counts set-systems, covering A003465, connected A323818.

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y Cf. A072639, A309326, A326031, A326702, A326753, A355739, A355741, A355744, A367771.

%K nonn

%O 0,5

%A _Gus Wiseman_, Dec 17 2023