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%I #17 May 23 2024 00:29:18
%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 1,1,3,1,1,0,1,1,0,0,1,0,0,0,3,1,1,0,2,0,0,0,1,0,0,0,0,0,0,0,3,2,2,1,
%U 4,1,1,0,2,1,1,0,2,0,0,0,4,1,2,0,3,0,0,0
%N Number of ways to choose a sequence of different binary indices, one 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}.
%H John Tyler Rascoe, <a href="/A367905/b367905.txt">Table of n, a(n) for n = 0..16384</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e 352 has binary indices of binary indices {{2,3},{1,2,3},{1,4}}, and there are six possible choices (2,1,4), (2,3,1), (2,3,4), (3,1,4), (3,2,1), (3,2,4), so a(352) = 6.
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
%t Table[Length[Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]],{n,0,100}]
%o (Python)
%o from itertools import count, islice, product
%o def bin_i(n): #binary indices
%o return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o def a_gen(): #generator of terms
%o for n in count(0):
%o c = 0
%o for j in list(product(*[bin_i(k) for k in bin_i(n)])):
%o if len(set(j)) == len(j):
%o c += 1
%o yield c
%o A367905_list = list(islice(a_gen(), 90)) # _John Tyler Rascoe_, May 22 2024
%Y A version for multisets is A367771, see A355529, A355740, A355744, A355745.
%Y Positions of positive terms are A367906.
%Y Positions of zeros are A367907.
%Y Positions of ones are A367908.
%Y Positions of terms > 1 are A367909.
%Y Positions of first appearances are A367910, sorted A367911.
%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. A000612, A055621, A072639, A309326, A326031, A326675, A326702, A326753, A367902, A367903, A367904, A367912.
%Y BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers), A326783 (uniform), A326784 (regular), A326788 (simple), A330217 (achiral).
%K nonn,base
%O 0,5
%A _Gus Wiseman_, Dec 10 2023
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