A368532 - OEIS

%I #5 Dec 29 2023 10:56:46

%S 7,25,30,42,45,51,53,54,60,75,77,78,83,85,86,90,92,99,101,102,105,108,

%T 113,114,116,120,385,390,408,428,434,436,458,460,466,468,482,484,488,

%U 496,642,645,668,680,689,692,713,716,721,724,728,737,740,752,771,773

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

%C 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 A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.

%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.

%e The terms the corresponding set-systems begin:

%e    7: {{1},{2},{1,2}}

%e   25: {{1},{3},{1,3}}

%e   30: {{2},{1,2},{3},{1,3}}

%e   42: {{2},{3},{2,3}}

%e   45: {{1},{1,2},{3},{2,3}}

%e   51: {{1},{2},{1,3},{2,3}}

%e   53: {{1},{1,2},{1,3},{2,3}}

%e   54: {{2},{1,2},{1,3},{2,3}}

%e   60: {{1,2},{3},{1,3},{2,3}}

%e   75: {{1},{2},{3},{1,2,3}}

%e   77: {{1},{1,2},{3},{1,2,3}}

%e   78: {{2},{1,2},{3},{1,2,3}}

%e   83: {{1},{2},{1,3},{1,2,3}}

%e   85: {{1},{1,2},{1,3},{1,2,3}}

%e   86: {{2},{1,2},{1,3},{1,2,3}}

%e   90: {{2},{3},{1,3},{1,2,3}}

%e   92: {{1,2},{3},{1,3},{1,2,3}}

%e   99: {{1},{2},{2,3},{1,2,3}}

%t vmin[y_]:=Select[y,Function[s,Select[DeleteCases[y,s], SubsetQ[bpe[s],bpe[#]]&]=={}]];

%t Select[Range[100],Select[Tuples[bpe/@bpe[#]] ,UnsameQ@@#&]=={}&]//vmin

%Y The version for MM-numbers of multiset partitions is A368187.

%Y A000110 counts set partitions.

%Y A003465 counts covering set-systems, unlabeled A055621.

%Y A058891 counts set-systems, unlabeled A000612, connected A323818.

%Y A283877 counts non-isomorphic set-systems, connected A300913.

%Y Cf. A007716, A134964, A355529, A367905, A367907.

%Y Cf. A140637, A367867, A367903, A368094, A368097, A368413.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 29 2023