A327053 - OEIS

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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).

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.)