A395250 Number of metrizable convexities on a set of n elements.

1, 1, 4, 74, 8508, 6460153

Offset

1,3

Comments

A convexity C on a set X is a collection of subsets of X such that {} and X are in C, C is closed under arbitrary intersection, and C is closed under directed union.

A convexity C is said to be metrizable if there exists a metric d on X so that for all subsets A of X, we have that A is convex iff z in A for all x,y in A and z in X with d(x,z)+d(z,y)=d(x,y) (see van de Vel).

Note that every convexity C on X induced by a metric d comes from a betweenness relation B defined by B(x,z,y)<=>d(x,z)+d(z,y)=d(x,y) for all x,y,z in X. Specifically, A is in C iff z in A whenever x,y in A, z in X, and B(x,z,y).

The Python program in the Links enumerates a(n) by first computing metrizable betweenness relations on a set of n elements (see A395237). Interestingly, a(n)=A395237(n) for n<=4 (distinct betweenness relations induce distinct convexities), but a(n)<A395237(n) for n>4.

Every metrizable convexity on a set X is an interval convexity (see A364656), with intervals given by metric segments, so a(n)<=A364656(n).

References

M. L. J. van de Vel, Theory of Convex Structures, North-Holland Mathematical Library, vol. 50, North-Holland, Amsterdam, 1993.

Links

Table of n, a(n) for n=1..6.

Tian Vlasic, Python Program

Example

a(3)=4 as there are 4 metrizable convexities on X={a,b,c}:
  {{},{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}},
  {{},{a},{b},{c},{a,b},{a,c},{a,b,c}},
  {{},{a},{b},{c},{a,b},{b,c},{a,b,c}}, and
  {{},{a},{b},{c},{a,c},{b,c},{a,b,c}}.

Crossrefs

Cf. A395237, A364656, A102894.

Sequence in context: A358568 A139112 A100865 * A395237 A007157 A000857

Adjacent sequences: A395246 A395247 A395248 * A395251 A395252 A395253

Keyword

nonn,more

Author

Tian Vlasic, May 20 2026

Extensions

a(6) from Robert P. P. McKone, May 24 2026

Status

approved