Enumerative Combinatorics of Homogeneous Linear Orderings

TL;DR

This paper derives explicit formulas and bounds for counting countable homogeneous linear orderings with various homogeneity notions, revealing finiteness and asymptotic properties.

Contribution

It introduces formulas and bounds for enumerating countable homogeneous linear orderings under different homogeneity conditions.

Findings

Explicit formulas for counting homogeneous linear orderings.

Asymptotic bounds for the number of such orderings.

Proof of finiteness for certain classes of orderings.

Abstract

We count the number of countable homogeneous colored linear orderings in $k$ colors. Relatedly, we count the number of countable $C_{n,m}$-homogeneous linear orderings. $C_{n,m}$-homogeneity is a strong homogeneity notion that approximates $sp-$homogeneity, a notion recently uncovered in [2] to have important computability theoretic properties. Explicit formulas are derived for both of the quantities in question, along with asymptotic bounds. The objects being counted are generally infinite, and it is not obvious that there are even only finitely many. This fact, along with the more precise counting, is demonstrated by corresponding the linear orderings with finite objects.