A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

0, 0, 1, 23, 1105

Offset

0,4

Comments

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=0..4.

Wikipedia, Axiom of choice.

Formula

A367903(n) + A367904(n) + a(n) = A058891(n).

Example

Non-isomorphic representatives of the a(3) = 23 set-systems:
  {{1,2}}
  {{1,2,3}}
  {{1},{2,3}}
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#], UnsameQ@@#&]]>1&]], {n, 0, 3}]

Crossrefs

For at least one choice we have A367902.

For no choices we have A367903, no singletons A367769, ranks A367907.

For a unique choice we have A367904, ranks A367908.

These set-systems have ranks A367909.

A000372 counts antichains, covering A006126, nonempty A014466.

A003465 counts covering set-systems, unlabeled A055621.

A058891 counts set-systems, unlabeled A000612.

Cf. A059201, A102896, A133686, A283877, A306445, A323818, A355741, A367770, A367862, A367869, A367901, A367905.

Sequence in context: A126741 A273477 A202656 * A340487 A321569 A061063

Adjacent sequences: A367769 A367770 A367771 * A367773 A367774 A367775

Keyword

nonn,more

Author

Gus Wiseman, Dec 12 2023

Status

approved