Enumerative Combinatorics of Homogeneous Linear Orderings

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.