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%I #7 Dec 12 2023 08:41:28
%S 4,12,16,18,20,32,33,36,48,52,64,65,66,68,72,76,80,82,84,96,97,100,
%T 112,132,140,144,146,148,160,161,164,176,180,192,193,194,196,200,204,
%U 208,210,212,224,225,228,240,256,258,260,264,266,268,272,274,276,288
%N Numbers n such that there is more than one way to choose a different binary index of each binary index of n.
%C Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in more than one way.
%C A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary digits (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.
%C The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%F A367907 U A367908 U A367909 = A000027.
%e The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in only one way (1,2,3), so 21 is not in the sequence.
%e The terms together with the corresponding set-systems begin:
%e 4: {{1,2}}
%e 12: {{1,2},{3}}
%e 16: {{1,3}}
%e 18: {{2},{1,3}}
%e 20: {{1,2},{1,3}}
%e 32: {{2,3}}
%e 33: {{1},{2,3}}
%e 36: {{1,2},{2,3}}
%e 48: {{1,3},{2,3}}
%e 52: {{1,2},{1,3},{2,3}}
%e 64: {{1,2,3}}
%e 65: {{1},{1,2,3}}
%e 66: {{2},{1,2,3}}
%e 68: {{1,2},{1,2,3}}
%e 72: {{3},{1,2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[100], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]>1&]
%Y These set-systems are counted by A367772.
%Y Positions of terms > 1 in A367905, firsts A367910, sorted firsts A367911.
%Y If there is at least one choice we get A367906, counted by A367902.
%Y If there are no choices we get A367907, counted by A367903.
%Y If there is one unique choice we get A367908, counted by A367904.
%Y A048793 lists binary indices, length A000120, reverse A272020, 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 A326031 gives weight of the set-system with BII-number n.
%Y A368098 counts unlabeled multiset partitions per axiom, complement A368097.
%Y Cf. A000612, A055621, A072639, A309326, A326702, A326753, A355529, A368100.
%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
%O 1,1
%A _Gus Wiseman_, Dec 11 2023
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