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Number of strict interval closure operators on a set of n elements.
1

%I #32 May 14 2024 07:06:22

%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 on a set X is strict if the empty set is closed; it is an interval if for every subset S of X, 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