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A327038 Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting). 8
1, 2, 6, 34, 1020, 1188106 (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..5.

FORMULA

Binomial transform of A327037.

EXAMPLE

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

  {}  {}     {}

      {{1}}  {{1}}

             {{2}}

             {{1,2}}

             {{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: A317080 A075272 A224913 * A228931 A101262 A135965

Adjacent sequences:  A327035 A327036 A327037 * A327039 A327040 A327041

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 17 2019

STATUS

approved

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Last modified January 28 22:30 EST 2020. Contains 331328 sequences. (Running on oeis4.)