A381471 -id:A381471 - cp's OEIS frontend

A380571 Number of Dynkin systems on [n].

1, 1, 2, 5, 19, 137, 3708, 1506404, 230328505024

Offset: 0

Peter J. Taylor, Feb 24 2025

A Dynkin system on a set S is a subset of the power set of S which contains the empty set, is closed under complements in S, and is closed under union of disjoint sets.

			The a(3) = 5 systems are:
  {{}, {1,2,3}}
  {{}, {1}, {2,3}, {1,2,3}}
  {{}, {2}, {1,3}, {1,2,3}}
  {{}, {3}, {1,2}, {1,2,3}}
  {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
The a(4) = 19 systems are 15 sigma-algebras counted by A000110(4) and 4 other systems:
  {{}, {1,2,3,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}
  {{}, {1,2,3,4}, {1,2}, {1,3}, {2,4}, {3,4}}
  {{}, {1,2,3,4}, {1,2}, {1,4}, {2,3}, {3,4}}
  {{}, {1,2,3,4}, {1,3}, {1,4}, {2,3}, {2,4}}

Martin Rubey and Peter Taylor, What is the number of finite Dynkin systems?, MathOverflow.
Wikipedia, Dynkin system

Cf. A000110, A102894, A381471 (unlabeled case).

a(n) >= A000110(n).

A381472 Number of unlabeled set systems on n vertices which are closed under union of disjoint sets.

Original entry on oeis.org

1, 2, 5, 22, 345, 152589

Offset: 0

Andrew Howroyd, Mar 02 2025

A set system is a finite set of finite nonempty sets.

a(n) is also the number of non-isomorphic sets of subsets of an n-set which are closed under union of disjoint sets and which include the empty set.

			Non-isomorphic representatives of the a(2) = 5 set-systems:
  {}
  {{1}}
  {{1,2}}
  {{1},{1,2}}
  {{1},{2},{1,2}}
The a(3) = 22 non-isomorphic set-systems include A193674(3) = 19 set-systems that are closed under union and 3 additional set-systems which do not include {1,2,3}:
  {{1,2},{1,3}}
  {{1},{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}

Cf. A000612 (set-systems), A193674 (closed under union of sets), A381471, A381575 (labeled case).

cp's OEIS Frontend

A380571 Number of Dynkin systems on [n].

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