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A364656 Number of strict interval closure operators on a set of n elements.

%I A364656 #32 May 14 2024 07:06:22
%S A364656 1,1,4,45,2062,589602,1553173541
%N A364656 Number of strict interval closure operators on a set of n elements.
%C A364656 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 A364656 a(n) is also the number of interval convexities on a set of n elements (see Chepoi).
%D A364656 G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.
%H A364656 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 A364656 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 A364656 Wikipedia, <a href="https://en.wikipedia.org/wiki/Closure_operator">Closure operator</a>
%e A364656 The a(3) = 45 set-systems are the following ({} and {1,2,3} not shown).
%e A364656     {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}
%e A364656     {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}
%e A364656     {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}
%e A364656     {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}
%e A364656     {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}
%e A364656     {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}
%e A364656           {2}{12}  {1}{3}{23}   {1}{3}{13}{23}
%e A364656           {2}{13}  {2}{3}{12}   {2}{3}{12}{23}
%e A364656           {2}{23}  {2}{3}{13}   {2}{3}{13}{23}
%e A364656           {3}{12}  {2}{3}{23}
%e A364656           {3}{13}  {1}{12}{13}
%e A364656           {3}{23}  {2}{12}{23}
%e A364656                    {3}{13}{23}
%t A364656 Table[With[{closure = {X, set} |->
%t A364656       Intersection @@ Select[X, SubsetQ[#, set] &]},
%t A364656    Select[
%t A364656     Select[
%t A364656      Join[{{}, Range@n}, #] & /@ Subsets@Subsets[Range@n, {1, n - 1}],
%t A364656       SubsetQ[#, Intersection @@@ Subsets[#, {2}]] &],
%t A364656     X |->
%t A364656      AllTrue[Complement[Subsets@Range@n, X],
%t A364656       S |-> \[Not]
%t A364656         AllTrue[Subsets[S, {1, 2}], SubsetQ[S, closure[X, #]] &]]]] //
%t A364656    Length, {n, 4}]
%Y A364656 Cf. A334255, A358144, A358152, A356544.
%K A364656 nonn,hard,more
%O A364656 0,3
%A A364656 _Tian Vlasic_, Jul 31 2023
%E A364656 New offset and a(5)-a(6) from _Dmitry I. Ignatov_, Nov 14 2023