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Except for a(0) = 1, this is also the number of antichains of nonempty sets covering n vertices (A006126). There are two antichains of size zero, namely {} and {{}}, while there is only one simplicial complex, namely {}. The unlabeled case is A261005. The non-covering case is A014466.

Also number of pure (= irreducible) group-testing histories of n items - A. Boneh, Mar 31 2000

Also number of antichain covers of an unlabeled n-set, so a(n) equals first differences of A003182. - Vladeta Jovovic, Goran Kilibarda, Aug 18 2000

Also number of inequivalent (under permutation of variables) nondegenerate monotone Boolean functions of n variables. We say h and g (functions of n variables) are equivalent if there exists a permutation p of S_n such that hp=g. E.g., a(3)=5 because xyz, xy+xz+yz, x+yz+xyz, xy+xz+xyz, x+y+z+xy+xz+yz+xyz are 5 inequivalent nondegenerate monotone Boolean functions that generate (by permutation of variables) the other 4. For example, y+xz+xyz can be obtained from x+yz+xyz by exchanging x and y. - Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 16 2006

The non-spanning/covering case is A003182. The labeled case is A006126. - Gus Wiseman, Feb 20 2019

Sometimes called Hosten-Morris numbers (or HM numbers).

Also the number of simplicial complexes on the set {1, ..., n-1} such that no pair of faces covers all of {1, ..., n-1}. [Miller-Sturmfels] - N. J. A. Sloane, Feb 18 2008

Also the maximal number of generators of a neighborly monomial ideal in n variables. [Miller-Sturmfels]. - N. J. A. Sloane, Feb 18 2008

Also the number of intersecting antichains on a labeled (n-1)-set or (n-1)-variable Boolean functions in the Post class F(7,2). Cf. A059090. - Vladeta Jovovic, Goran Kilibarda, Dec 28 2000

Also the number of nondominated coteries on n members. - Don Knuth, Sep 01 2005

The number of maximal families of intersecting subsets of an n-element set. - Bridget Tenner, Nov 16 2006

Rivière gives a(n) for n <= 5. - N. J. A. Sloane, May 12 2012

Only the non-singleton edges are required to form an antichain.

Number of non-degenerate unate Boolean functions of n or fewer variables. - Aniruddha Biswas, May 11 2024

From Gus Wiseman, Jul 03 2019: (Start)

Also the number of antichains covering n vertices and having empty intersection (meaning there is no vertex in common to all the edges). For example, the a(3) = 5 antichains are:

{{3},{1,2}}

{{2},{1,3}}

{{1},{2,3}}

{{1},{2},{3}}

{{1,2},{1,3},{2,3}}

(End)

Equivalently, the number of elements of the free distributive lattice with n generators.

A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}.

The count of antichains excludes the empty antichain which contains no subsets and the antichain consisting of only the empty set.

The number of continuous functions f : R^n->R with f(x_1,..,x_n) in {x_1,..,x_n}. - Jan Fricke, Feb 12 2004

From Robert FERREOL, Mar 23 2009: (Start)

a(n) is also the number of reduced normal conjunctive forms with n variables without negation.

For example the 18 forms for n=3 are :

a

b

c

a or b

a or c

b or c

a or b or c

a and b

a and c

b and c

a and (b or c)

b and (a or c)

c and (a or b)

(a or b) and (a or c)

(b or a) and (b or c)

(c or a) and (c or b)

a and b and c

(a or b) and (a or c) and (b or c)

(End)

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In a chain of sets, every edge is a subset or superset of every other edge.

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.

Includes all set-systems with more edges than covered vertices, but this condition is not sufficient.