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A364656
Number of strict interval closure operators on a set of n elements.
1
1, 1, 4, 45, 2062, 589602, 1553173541
OFFSET
0,3
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).
REFERENCES
G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.
EXAMPLE
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}
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}]
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Tian Vlasic, Jul 31 2023
EXTENSIONS
New offset and a(5)-a(6) from Dmitry I. Ignatov, Nov 14 2023
STATUS
approved