A102894 -id:A102894 - 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).

A235604 Number of equivalence classes of lattices of subsets of the power set 2^[n].

Original entry on oeis.org

1, 1, 1, 4, 50, 7443, 95239971

Offset: 0

N. J. A. Sloane, Jan 21 2014

This is also the number of inequivalent atomic lattices on n atoms or inequivalent strict closure systems under T1 separation axiom on n elements. - Dmitry I. Ignatov, Sep 27 2022

Donald M. Davis, Enumerating lattices of subsets, arXiv preprint arXiv:1311.6664 [math.CO], 2013.
Dmitry I. Ignatov, On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom, arXiv:2209.12256 [cs.DM], 2022.

The number of inequivalent closure operators on a set of n elements where all singletons are closed is given in A355517.
The number of all strict closure operators is given in A102894.
For T_1 closure operators, see A334254.

a(5) from Andrew Weimholt, Jan 27 2014

a(6) from Dmitry I. Ignatov, Sep 27 2022

A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

Original entry on oeis.org

0, 0, 0, 2, 27, 3133, 5777931

Offset: 1

Gunnar Brinkmann, Feb 05 2018

If there is a counterexample to the union-closed set conjecture, it is a sparse union-closed set.

G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.

Cf. A108798, A102894, A193674, A102896.

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A380571 Number of Dynkin systems on [n].

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A235604 Number of equivalence classes of lattices of subsets of the power set 2^[n].

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A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

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