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%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
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