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 A059201 Number of T_0-covers of a labeled n-set. 64
 1, 1, 4, 96, 31692, 2147001636, 9223371991763269704, 170141183460469231473432887375376674952, 57896044618658097711785492504343953920509909728243389682424010192567186540224 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point. From Gus Wiseman, Aug 13 2019: (Start) A set-system is a finite set of finite nonempty sets. 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}}. The T_0 condition means that the dual is strict (no repeated edges). For example, the a(2) = 4 covers are:   {{1},{2}}   {{1},{1,2}}   {{2},{1,2}}   {{1},{2},{1,2}} (End) LINKS G. C. Greubel, Table of n, a(n) for n = 0..11 Vladeta Jovovic, T_0-covers of a labeled 3-set FORMULA a(n) = Sum_{i=0..n+1} stirling1(n+1, i)*2^(2^(i-1)-1). a(n) = Sum_{m=0..2^n-1} A059202(n,m). Inverse binomial transform of A326940 and exponential transform of A326948. - Gus Wiseman, Aug 13 2019 MATHEMATICA Table[Sum[StirlingS1[n + 1, k]*2^(2^(k - 1) - 1), {k, 0, n + 1}], {n, 0, 5}] (* G. C. Greubel, Dec 28 2016 *) dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}]; Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&]], {n, 0, 3}] (* Gus Wiseman, Aug 13 2019 *) CROSSREFS Row sums of A059202. Cf. A059203, A059084, A059085, A059086, A059087, A059088, A059089. Covering set-systems are A003465. The unlabeled version is A319637. The version with empty edges allowed is A326939. The non-covering version is A326940. BII-numbers of T_0 set-systems are A326947. The same with connected instead of covering is A326948. The T_1 version is A326961. Cf. A245567, A316978, A319559, A319564, A323818, A326941, A326946, A326970. Sequence in context: A181335 A098695 A307934 * A323818 A027638 A309483 Adjacent sequences:  A059198 A059199 A059200 * A059202 A059203 A059204 KEYWORD easy,nonn AUTHOR Vladeta Jovovic, Goran Kilibarda, Jan 16 2001 STATUS approved

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Last modified February 23 15:54 EST 2020. Contains 332170 sequences. (Running on oeis4.)