cp's OEIS Frontend

A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.

%I A334254 #27 Feb 16 2025 08:34:00
%S A334254 1,2,1,8,545,702525,66960965307
%N A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.
%C A334254 The T_1 axiom states that all singleton sets {x} are closed.
%C A334254 For n>1, this property implies strictness (meaning that the empty set is closed).
%H A334254 Dmitry I. Ignatov, <a href="http://arxiv.org/abs/2209.12256">On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom</a>, arXiv:2209.12256 [cs.DM], 2022.
%H A334254 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 A334254 Dmitry I. Ignatov, <a href="/A334254/a334254.pdf">PDF of the supporting iPython notebook</a>
%H A334254 S. Mapes, <a href="https://www3.nd.edu/~smapes1/FALRMI.pdf">Finite atomic lattices and resolutions of monomial ideals</a>, J. Algebra,  379 (2013), 259-276.
%H A334254 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SeparationAxioms.html">Separation Axioms</a>
%H A334254 Wikipedia, <a href="http://en.wikipedia.org/wiki/Separation_axiom">Separation Axiom</a>
%e A334254 The a(3) = 8 set-systems of closed sets:
%e A334254   {{1,2,3},{1},{2},{3},{}}
%e A334254   {{1,2,3},{1,2},{1},{2},{3},{}}
%e A334254   {{1,2,3},{1,3},{1},{2},{3},{}}
%e A334254   {{1,2,3},{2,3},{1},{2},{3},{}}
%e A334254   {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
%e A334254   {{1,2,3},{1,2},{2,3},{1},{2},{3},{}}
%e A334254   {{1,2,3},{1,3},{2,3},{1},{2},{3},{}}
%e A334254   {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}
%Y A334254 The number of all closure operators is given in A102896.
%Y A334254 For T_0 closure operators, see A334252.
%Y A334254 For strict T_1 closure operators, see A334255, the only difference is a(1).
%Y A334254 Cf. A326960, A326961, A326979.
%K A334254 nonn,more,hard
%O A334254 0,2
%A A334254 _Joshua Moerman_, Apr 20 2020
%E A334254 a(6) from _Dmitry I. Ignatov_, Jul 03 2022