A327038 - OEIS

A327038

Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

1, 2, 6, 34, 1020, 1188106, 909149847892, 291200434288840793135801

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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}}. This sequence counts pairwise intersecting set-systems that are cointersecting, meaning their dual is pairwise intersecting.

LINKS

Table of n, a(n) for n=0..7.

FORMULA

Binomial transform of A327037.

EXAMPLE

The a(0) = 1 through a(2) = 6 set-systems:

{} {} {}

{{1},{1,2}}

{{2},{1,2}}

The a(3) = 34 set-systems:

{} {{1}} {{1}{12}} {{1}{12}{123}} {{1}{12}{13}{123}}

{{2}} {{1}{13}} {{1}{13}{123}} {{2}{12}{23}{123}}

{{3}} {{2}{12}} {{12}{13}{23}} {{3}{13}{23}{123}}

{{12}} {{2}{23}} {{2}{12}{123}} {{12}{13}{23}{123}}

{{13}} {{3}{13}} {{2}{23}{123}}

{{23}} {{3}{23}} {{3}{13}{123}}

{{123}} {{1}{123}} {{3}{23}{123}}

{{2}{123}} {{12}{13}{123}}

{{3}{123}} {{12}{23}{123}}

{{12}{123}} {{13}{23}{123}}

{{13}{123}}

{{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]=={}&], stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 4}]

CROSSREFS

Intersecting set-systems are A051185.

The unlabeled multiset partition version is A319765.

The BII-numbers of these set-systems are A326912.

The covering case is A327037.

Cointersecting set-systems are A327039.

The case where the dual is strict is A327040.

Cf. A058891, A319767, A319774, A326854, A327052.

Sequence in context: A075272 A353536 A224913 * A228931 A101262 A135965

Adjacent sequences: A327035 A327036 A327037 * A327039 A327040 A327041