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A059201
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Number of T_0-covers of a labeled n-set.
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89
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1, 1, 4, 96, 31692, 2147001636, 9223371991763269704, 170141183460469231473432887375376674952, 57896044618658097711785492504343953920509909728243389682424010192567186540224
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OFFSET
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0,3
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COMMENTS
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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.
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)
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LINKS
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FORMULA
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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).
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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