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

1,1

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.

Links

Table of n, a(n) for n=1..56.

Example

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}}

Mathematica

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

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.

Sequence in context: A272366 A327107 A075926 * A065660 A100496 A110081

Adjacent sequences: A368529 A368530 A368531 * A368533 A368534 A368535

Keyword

nonn

Author

Gus Wiseman, Dec 29 2023

Status

approved