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A367905 Number of ways to choose a sequence of different binary indices, one of each binary index of n.

%I A367905 #17 May 23 2024 00:29:18
%S A367905 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 A367905 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 A367905 4,1,1,0,2,1,1,0,2,0,0,0,4,1,2,0,3,0,0,0
%N A367905 Number of ways to choose a sequence of different binary indices, one of each binary index of n.
%C A367905 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 A367905 John Tyler Rascoe, <a href="/A367905/b367905.txt">Table of n, a(n) for n = 0..16384</a>
%H A367905 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A367905 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 A367905 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
%t A367905 Table[Length[Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]],{n,0,100}]
%o A367905 (Python)
%o A367905 from itertools import count, islice, product
%o A367905 def bin_i(n): #binary indices
%o A367905     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A367905 def a_gen(): #generator of terms
%o A367905     for n in count(0):
%o A367905         c = 0
%o A367905         for j in list(product(*[bin_i(k) for k in bin_i(n)])):
%o A367905             if len(set(j)) == len(j):
%o A367905                 c += 1
%o A367905         yield c
%o A367905 A367905_list = list(islice(a_gen(), 90)) # _John Tyler Rascoe_, May 22 2024
%Y A367905 A version for multisets is A367771, see A355529, A355740, A355744, A355745.
%Y A367905 Positions of positive terms are A367906.
%Y A367905 Positions of zeros are A367907.
%Y A367905 Positions of ones are A367908.
%Y A367905 Positions of terms > 1 are A367909.
%Y A367905 Positions of first appearances are A367910, sorted A367911.
%Y A367905 A048793 lists binary indices, length A000120, sum A029931.
%Y A367905 A058891 counts set-systems, covering A003465, connected A323818.
%Y A367905 A070939 gives length of binary expansion.
%Y A367905 A096111 gives product of binary indices.
%Y A367905 Cf. A000612, A055621, A072639, A309326, A326031, A326675, A326702, A326753, A367902, A367903, A367904, A367912.
%Y A367905 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 A367905 nonn,base
%O A367905 0,5
%A A367905 _Gus Wiseman_, Dec 10 2023