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

1, 1, 4, 45, 2062, 589602, 1553173541

Offset: 0

Author

Tian Vlasic, Jul 31 2023

Keywords

• nonn
• hard
• more

Comments

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.

a(n) is also the number of interval convexities on a set of n elements (see Chepoi).

Examples

			The a(3) = 45 set-systems are the following ({} and {1,2,3} not shown).
    {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}
    {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}
    {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}
    {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}
    {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}
    {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}
          {2}{12}  {1}{3}{23}   {1}{3}{13}{23}
          {2}{13}  {2}{3}{12}   {2}{3}{12}{23}
          {2}{23}  {2}{3}{13}   {2}{3}{13}{23}
          {3}{12}  {2}{3}{23}
          {3}{13}  {1}{12}{13}
          {3}{23}  {2}{12}{23}
                   {3}{13}{23}
		

References

• G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.

Links

• Victor Chepoi, Separation of Two Convex Sets in Convexity Structures
• Dmitry I. Ignatov, Supporting iPython code for counting strict interval closure operators up to n=6, Github repository
• Wikipedia, Closure operator

Crossrefs

Cf. A334255, A358144, A358152, A356544.

Programs

Mathematica

Table[With[{closure = {X, set} |->
      Intersection @@ Select[X, SubsetQ[#, set] &]},
   Select[
    Select[
     Join[{{}, Range@n}, #] & /@ Subsets@Subsets[Range@n, {1, n - 1}],
      SubsetQ[#, Intersection @@@ Subsets[#, {2}]] &],
    X |->
     AllTrue[Complement[Subsets@Range@n, X],
      S |-> \[Not]
        AllTrue[Subsets[S, {1, 2}], SubsetQ[S, closure[X, #]] &]]]] //
   Length, {n, 4}]

Extensions

New offset and a(5)-a(6) from Dmitry I. Ignatov, Nov 14 2023