A327020 - OEIS

A327020

Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

1, 1, 1, 2, 17, 1451, 3741198

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OFFSET

0,4

COMMENTS

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges, The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of sets, none of which is a subset of any other. This sequence counts antichains with union {1..n} whose dual is pairwise intersecting.

LINKS

Table of n, a(n) for n=0..6.

FORMULA

Inverse binomial transform of A327057.

EXAMPLE

The a(0) = 1 through a(4) = 17 antichains:

{} {{1}} {{12}} {{123}} {{1234}}

{{12}{13}{23}} {{12}{134}{234}}

{{13}{124}{234}}

{{14}{123}{234}}

{{23}{124}{134}}

{{24}{123}{134}}

{{34}{123}{124}}

{{123}{124}{134}}

{{123}{124}{234}}

{{123}{134}{234}}

{{124}{134}{234}}

{{12}{13}{14}{234}}

{{12}{23}{24}{134}}

{{13}{23}{34}{124}}

{{14}{24}{34}{123}}

{{123}{124}{134}{234}}

{{12}{13}{14}{23}{24}{34}}

MATHEMATICA

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];

stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];

Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], Union@@#==Range[n]&&stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 4}]

CROSSREFS

Covering, intersecting antichains are A305844.

Covering, T1 antichains are A319639.

Cointersecting set-systems are A327039.

Covering, cointersecting set-systems are A327040.

Covering, cointersecting set-systems are A327051.

The non-covering version is A327057.

Covering, intersecting, T1 set-systems are A327058.

Unlabeled cointersecting antichains of multisets are A327060.

Cf. A003465, A305843, A319765, A326853, A327037, A327038, A327053.