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%I #9 Mar 10 2024 15:12:07
%S 1,2,4,6,12,19,30,45,90,147,230,343,504,716,994,1352,2704,4349,6469
%N Number of subsets of {1..n} such that a unique set can be obtained by choosing 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>.
%F a(2^n - 1) = A370818(n).
%e The set {3,4} has binary indices {{1,2},{3}}, with two choices {1,3}, {2,3}, so is not counted under a(4).
%e The a(0) = 1 through a(5) = 19 subsets:
%e {} {} {} {} {} {}
%e {1} {1} {1} {1} {1}
%e {2} {2} {2} {2}
%e {1,2} {1,2} {4} {4}
%e {1,3} {1,2} {1,2}
%e {2,3} {1,3} {1,3}
%e {1,4} {1,4}
%e {2,3} {1,5}
%e {2,4} {2,3}
%e {1,2,4} {2,4}
%e {1,3,4} {4,5}
%e {2,3,4} {1,2,4}
%e {1,2,5}
%e {1,3,4}
%e {1,3,5}
%e {2,3,4}
%e {2,3,5}
%e {2,4,5}
%e {3,4,5}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Table[Length[Select[Subsets[Range[n]],Length[Union[Sort /@ Select[Tuples[bpe/@#],UnsameQ@@#&]]]==1&]],{n,0,10}]
%Y Set systems of this type are counted by A367904, ranks A367908.
%Y A version for MM-numbers of multisets is A368101.
%Y For prime indices we have A370584.
%Y This is the unique version of A370636, complement A370637.
%Y The maximal case is A370640, differences A370641.
%Y Factorizations of this type are counted by A370645.
%Y The case A370818 is the restriction to A000225.
%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 Cf. A133686, A134964, A326031, A326702, A367772, A367867, A367905, A367909, A367912, A368109.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Mar 09 2024
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