cp's OEIS Frontend

A368532 Minimal numbers whose binary indices of binary indices contradict a strict version of the axiom of choice.

%I A368532 #5 Dec 29 2023 10:56:46
%S A368532 7,25,30,42,45,51,53,54,60,75,77,78,83,85,86,90,92,99,101,102,105,108,
%T A368532 113,114,116,120,385,390,408,428,434,436,458,460,466,468,482,484,488,
%U A368532 496,642,645,668,680,689,692,713,716,721,724,728,737,740,752,771,773
%N A368532 Minimal numbers whose binary indices of binary indices contradict a strict version of the axiom of choice.
%C A368532 Minimality is relative to the ordering where x < y means the binary indices of x are a subset of those of y (a Boolean algebra).
%C A368532 A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.
%C A368532 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.
%e A368532 The terms the corresponding set-systems begin:
%e A368532    7: {{1},{2},{1,2}}
%e A368532   25: {{1},{3},{1,3}}
%e A368532   30: {{2},{1,2},{3},{1,3}}
%e A368532   42: {{2},{3},{2,3}}
%e A368532   45: {{1},{1,2},{3},{2,3}}
%e A368532   51: {{1},{2},{1,3},{2,3}}
%e A368532   53: {{1},{1,2},{1,3},{2,3}}
%e A368532   54: {{2},{1,2},{1,3},{2,3}}
%e A368532   60: {{1,2},{3},{1,3},{2,3}}
%e A368532   75: {{1},{2},{3},{1,2,3}}
%e A368532   77: {{1},{1,2},{3},{1,2,3}}
%e A368532   78: {{2},{1,2},{3},{1,2,3}}
%e A368532   83: {{1},{2},{1,3},{1,2,3}}
%e A368532   85: {{1},{1,2},{1,3},{1,2,3}}
%e A368532   86: {{2},{1,2},{1,3},{1,2,3}}
%e A368532   90: {{2},{3},{1,3},{1,2,3}}
%e A368532   92: {{1,2},{3},{1,3},{1,2,3}}
%e A368532   99: {{1},{2},{2,3},{1,2,3}}
%t A368532 vmin[y_]:=Select[y,Function[s,Select[DeleteCases[y,s], SubsetQ[bpe[s],bpe[#]]&]=={}]];
%t A368532 Select[Range[100],Select[Tuples[bpe/@bpe[#]] ,UnsameQ@@#&]=={}&]//vmin
%Y A368532 The version for MM-numbers of multiset partitions is A368187.
%Y A368532 A000110 counts set partitions.
%Y A368532 A003465 counts covering set-systems, unlabeled A055621.
%Y A368532 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368532 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368532 Cf. A007716, A134964, A355529, A367905, A367907.
%Y A368532 Cf. A140637, A367867, A367903, A368094, A368097, A368413.
%K A368532 nonn
%O A368532 1,1
%A A368532 _Gus Wiseman_, Dec 29 2023