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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, 2076948136, 9221293198653529144, 170141182628636920684331812494864430896 (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..7. 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 EXTENSIONS a(5)-a(7) from Christian Sievers, Feb 04 2024 STATUS approved

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Last modified May 30 02:13 EDT 2024. Contains 372957 sequences. (Running on oeis4.)