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%I #26 Nov 15 2023 08:04:54
%S 1,1,4,45,2062,589602,1553173541
%N Number of strict interval closure operators on a set of n elements.
%C A closure operator cl is strict if {} is closed; it is interval if for every set S, the statement that for all x,y in S, cl({x,y}) is a subset of S implies that S is closed.
%C a(n) is also the number of interval convexities on a set of n elements (see Chepoi).
%D G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.
%H Victor Chepoi, <a href="https://www.researchgate.net/publication/2407147_Separation_Of_Two_Convex_Sets_In_Convexity_Structures">Separation of Two Convex Sets in Convexity Structures</a>
%H Dmitry I. Ignatov, <a href="https://github.com/dimachine/StrictIntervalClosures/">Supporting iPython code for counting strict interval closure operators up to n=6</a>, Github repository
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Closure_operator">Closure operator</a>
%e The a(3) = 45 set-systems are the following ({} and {1,2,3} not shown).
%e {1} {1}{2} {1}{2}{3} {1}{2}{3}{12} {1}{2}{3}{12}{13}
%e {2} {1}{3} {1}{2}{12} {1}{2}{3}{13} {1}{2}{3}{12}{23}
%e {3} {2}{3} {1}{2}{13} {1}{2}{3}{23} {1}{2}{3}{13}{23}
%e {12} {1}{12} {1}{2}{23} {1}{2}{12}{13}
%e {13} {1}{13} {1}{3}{12} {1}{2}{12}{23}
%e {23} {1}{23} {1}{3}{13} {1}{3}{12}{13} {1}{2}{3}{12}{13}{23}
%e {2}{12} {1}{3}{23} {1}{3}{13}{23}
%e {2}{13} {2}{3}{12} {2}{3}{12}{23}
%e {2}{23} {2}{3}{13} {2}{3}{13}{23}
%e {3}{12} {2}{3}{23}
%e {3}{13} {1}{12}{13}
%e {3}{23} {2}{12}{23}
%e {3}{13}{23}
%t Table[With[{closure = {X, set} |->
%t Intersection @@ Select[X, SubsetQ[#, set] &]},
%t Select[
%t Select[
%t Join[{{}, Range@n}, #] & /@ Subsets@Subsets[Range@n, {1, n - 1}],
%t SubsetQ[#, Intersection @@@ Subsets[#, {2}]] &],
%t X |->
%t AllTrue[Complement[Subsets@Range@n, X],
%t S |-> \[Not]
%t AllTrue[Subsets[S, {1, 2}], SubsetQ[S, closure[X, #]] &]]]] //
%t Length, {n, 4}]
%Y Cf. A334255, A358144, A358152, A356544.
%K nonn,hard,more
%O 0,3
%A _Tian Vlasic_, Jul 31 2023
%E New offset and a(5)-a(6) from _Dmitry I. Ignatov_, Nov 14 2023
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