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A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.

%I A334255 #26 Feb 16 2025 08:34:00
%S A334255 1,1,1,8,545,702525,66960965307
%N A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.
%C A334255 The T_1 axiom states that all singleton sets {x} are closed.
%C A334255 A closure operator is strict if the empty set is closed.
%H A334255 Dmitry I. Ignatov, <a href="https://github.com/dimachine/ClosureSeparation/">Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom</a>, Github repository
%H A334255 Dmitry I. Ignatov, <a href="/A334255/a334255.ipynb.txt">Supporting iPython notebook</a>
%H A334255 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SeparationAxioms.html">Separation Axioms</a>
%H A334255 Wikipedia, <a href="http://en.wikipedia.org/wiki/Separation_axiom">Separation Axiom</a>
%e A334255 The a(3) = 8 set-systems of closed sets:
%e A334255   {{1,2,3},{1},{2},{3},{}}
%e A334255   {{1,2,3},{1,2},{1},{2},{3},{}}
%e A334255   {{1,2,3},{1,3},{1},{2},{3},{}}
%e A334255   {{1,2,3},{2,3},{1},{2},{3},{}}
%e A334255   {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
%e A334255   {{1,2,3},{1,2},{2,3},{1},{2},{3},{}}
%e A334255   {{1,2,3},{1,3},{2,3},{1},{2},{3},{}}
%e A334255   {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}
%t A334255 Table[Length[
%t A334255   Select[Subsets[Subsets[Range[n]]],
%t A334255    And[MemberQ[#, {}], MemberQ[#, Range[n]],
%t A334255      SubsetQ[#, Intersection @@@ Tuples[#, 2]],
%t A334255      SubsetQ[#, Map[{#} &, Range[n]]]] &]], {n, 0, 4}] (* _Tian Vlasic_, Jul 29 2022 *)
%Y A334255 The number of all strict closure operators is given in A102894.
%Y A334255 For all strict T_0 closure operators, see A334253.
%Y A334255 For T_1 closure operators, see A334254.
%Y A334255 Cf. A326960, A326961, A326979.
%K A334255 nonn,more
%O A334255 0,4
%A A334255 _Joshua Moerman_, Apr 24 2020
%E A334255 a(6) from _Dmitry I. Ignatov_, Jul 03 2022