A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

1, 1, 1, 3, 155

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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

Links

Table of n, a(n) for n=0..4.

Formula

Inverse binomial transform of A327059.

Example

The a(0) = 1 through a(3) = 3 set-systems:
  {}  {{1}}  {{12}}  {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

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}], Intersection[#1, #2]=={}&], Union@@#==Range[n]&&stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]

Crossrefs

Covering intersecting set-systems are A305843.

The BII-numbers of these set-systems are the intersection of A326910 and A326966.

Covering coantichains are A326970.

The non-covering version is A327059.

The unlabeled multiset partition version is A327060.

Cf. A006126, A051185, A059523, A305844, A319639, A326961, A326965, A326968, A327020, A327057.

Sequence in context: A156990 A075514 A344898 * A087306 A278877 A203682

Adjacent sequences: A327055 A327056 A327057 * A327059 A327060 A327061

Keyword

nonn,more

Author

Gus Wiseman, Aug 18 2019

Status

approved