A367901 Number of sets of subsets of {1..n} contradicting a strict version of the axiom of choice.

1, 2, 9, 195, 63765, 4294780073, 18446744073639513336, 340282366920938463463374607341656713953, 115792089237316195423570985008687907853269984665640564039457583610129753447747

Offset: 0

Author

Gus Wiseman, Dec 05 2023

Keywords

• nonn

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.

Examples

			The a(2) = 9 sets of sets:
  {{}}
  {{},{1}}
  {{},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}
		

Links

• Wikipedia, Axiom of choice.

Crossrefs

The version for simple graphs is A367867, covering A367868.

The complement is counted by A367902, no singletons A367770, ranks A367906.

The version without empty edges is A367903, ranks A367907.

For a unique choice (instead of none) we have A367904, ranks A367908.

A000372 counts antichains, covering A006126, nonempty A014466.

A003465 counts covering set-systems, unlabeled A055621.

A058891 counts set-systems, unlabeled A000612.

A059201 counts covering T_0 set-systems.

A323818 counts covering connected set-systems, unlabeled A323819.

A326031 gives weight of the set-system with BII-number n.

Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367769, A367862, A367905.

Programs

Mathematica

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

Formula

a(n) = 2^2^n - A367902(n). - Christian Sievers, Aug 01 2024

Extensions

a(5)-a(8) from Christian Sievers, Aug 01 2024