|
%I #7 Sep 06 2022 14:58:16
%S 1,2,1,4,50,7443,95239971
%N Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.
%C The T_1 axiom states that all singleton sets {x} are closed.
%C For n>1, this property implies strictness (meaning that the empty set is closed).
%H Dmitry I. Ignatov, <a href="https://github.com/dimachine/ClosureSeparation/">Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom</a>, Github repository
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeparationAxioms.html">Separation Axioms</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Separation_axiom">Separation Axiom</a>
%e a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}.
%e For a(2) = 1 the closure system is as follows: {{1,2},{1},{2},{}}.
%e The a(3) = 4 inequivalent set-systems of closed sets are:
%e {{1,2,3},{1},{2},{3},{}}
%e {{1,2,3},{1,2},{1},{2},{3},{}}
%e {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
%e {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}.
%Y The number of all closure operators is given in A102896, while A193674 contains the number of all nonisomorphic ones.
%Y For T_1 closure operators and their strict counterparts, see A334254 and A334255, respectively; the only difference is a(1).
%Y Cf. A326960, A326961, A326979.
%K nonn,hard,more
%O 0,2
%A _Dmitry I. Ignatov_, Jul 05 2022
|
