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%I #18 Aug 14 2019 19:35:11
%S 1,1,4,96,31692,2147001636,9223371991763269704,
%T 170141183460469231473432887375376674952,
%U 57896044618658097711785492504343953920509909728243389682424010192567186540224
%N Number of T_0-covers of a labeled n-set.
%C 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.
%C From _Gus Wiseman_, Aug 13 2019: (Start)
%C 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:
%C {{1},{2}}
%C {{1},{1,2}}
%C {{2},{1,2}}
%C {{1},{2},{1,2}}
%C (End)
%H G. C. Greubel, <a href="/A059201/b059201.txt">Table of n, a(n) for n = 0..11</a>
%H Vladeta Jovovic, <a href="/A059201/a059201.pdf">T_0-covers of a labeled 3-set</a>
%F a(n) = Sum_{i=0..n+1} stirling1(n+1, i)*2^(2^(i-1)-1).
%F a(n) = Sum_{m=0..2^n-1} A059202(n,m).
%F Inverse binomial transform of A326940 and exponential transform of A326948. - _Gus Wiseman_, Aug 13 2019
%t Table[Sum[StirlingS1[n + 1, k]*2^(2^(k - 1) - 1), {k, 0, n + 1}], {n,0,5}] (* _G. C. Greubel_, Dec 28 2016 *)
%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&]],{n,0,3}] (* _Gus Wiseman_, Aug 13 2019 *)
%Y Row sums of A059202.
%Y Cf. A059203, A059084, A059085, A059086, A059087, A059088, A059089.
%Y Covering set-systems are A003465.
%Y The unlabeled version is A319637.
%Y The version with empty edges allowed is A326939.
%Y The non-covering version is A326940.
%Y BII-numbers of T_0 set-systems are A326947.
%Y The same with connected instead of covering is A326948.
%Y The T_1 version is A326961.
%Y Cf. A245567, A316978, A319559, A319564, A323818, A326941, A326946, A326970.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Goran Kilibarda, Jan 16 2001
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