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

7, 25, 30, 42, 45, 51, 53, 54, 60, 75, 77, 78, 83, 85, 86, 90, 92, 99, 101, 102, 105, 108, 113, 114, 116, 120, 385, 390, 408, 428, 434, 436, 458, 460, 466, 468, 482, 484, 488, 496, 642, 645, 668, 680, 689, 692, 713, 716, 721, 724, 728, 737, 740, 752, 771, 773

Offset: 1

Author

Gus Wiseman, Dec 29 2023

Keywords

• nonn

Comments

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).

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

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.

Examples

			The terms the corresponding set-systems begin:
   7: {{1},{2},{1,2}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  75: {{1},{2},{3},{1,2,3}}
  77: {{1},{1,2},{3},{1,2,3}}
  78: {{2},{1,2},{3},{1,2,3}}
  83: {{1},{2},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  86: {{2},{1,2},{1,3},{1,2,3}}
  90: {{2},{3},{1,3},{1,2,3}}
  92: {{1,2},{3},{1,3},{1,2,3}}
  99: {{1},{2},{2,3},{1,2,3}}
		

Crossrefs

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

A000110 counts set partitions.

A003465 counts covering set-systems, unlabeled A055621.

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

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

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

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

Programs

Mathematica

vmin[y_]:=Select[y,Function[s,Select[DeleteCases[y,s], SubsetQ[bpe[s],bpe[#]]&]=={}]];
Select[Range[100],Select[Tuples[bpe/@bpe[#]] ,UnsameQ@@#&]=={}&]//vmin