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A327053 Number of T_0 (costrict) set-systems covering n vertices where every two vertices appear together in some edge (cointersecting). 6
1, 1, 3, 62, 24710 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 covering set-systems whose dual is strict and pairwise intersecting.

LINKS

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

FORMULA

Inverse binomial transform of A327052.

EXAMPLE

The a(1) = 1 through a(2) = 3 set-systems:

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

{{2},{1,2}}

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

The a(3) = 62 set-systems:

1 2 123 1 2 3 123 1 2 12 13 23 1 2 3 12 13 23 1 2 3 12 13 23 123

1 3 123 1 12 13 23 1 2 3 12 123 1 2 3 12 13 123

2 3 123 1 2 12 123 1 2 3 13 123 1 2 3 12 23 123

1 12 123 1 2 13 123 1 2 3 23 123 1 2 3 13 23 123

1 13 123 1 2 23 123 1 3 12 13 23 1 2 12 13 23 123

12 13 23 1 3 12 123 2 3 12 13 23 1 3 12 13 23 123

2 12 123 1 3 13 123 1 2 12 13 123 2 3 12 13 23 123

2 23 123 1 3 23 123 1 2 12 23 123

3 13 123 2 12 13 23 1 2 13 23 123

3 23 123 2 3 12 123 1 3 12 13 123

12 13 123 2 3 13 123 1 3 12 23 123

12 23 123 2 3 23 123 1 3 13 23 123

13 23 123 3 12 13 23 2 3 12 13 123

1 12 13 123 2 3 12 23 123

1 12 23 123 2 3 13 23 123

1 13 23 123 1 12 13 23 123

2 12 13 123 2 12 13 23 123

2 12 23 123 3 12 13 23 123

2 13 23 123

3 12 13 123

3 12 23 123

3 13 23 123

12 13 23 123

MATHEMATICA

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

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

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 3}]

CROSSREFS

The pairwise intersecting case is A319774.

The BII-numbers of these set-systems are the intersection of A326947 and A326853.

The non-T_0 version is A327040.

The non-covering version is A327052.

Cf. A003465, A305843, A319767, A326854, A327020, A327037, A327039.

Sequence in context: A104403 A301609 A144422 * A135444 A333041 A323726

Adjacent sequences: A327050 A327051 A327052 * A327054 A327055 A327056

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 18 2019

STATUS

approved

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Last modified January 30 19:10 EST 2023. Contains 359947 sequences. (Running on oeis4.)