Enumeration of Finite Distance Monoids

Building on the work of Gabriel Conant, we investigate the enumeration problems of finite distance monoids by applying the decomposition of Archimedean classes and studying their internal arithmetic progressions. Specifically, we first determine the exact value of $DM(n,2)$, which denotes the number of distance monoids on $n$ non-zero elements with Archimedean complexity $2$. This computation allows us to resolve a conjecture of Conant, establishing that the total number $DM(n)$ of distance monoids grows at least exponentially in $n$. Furthermore, we study the asymptotic behavior of $DM(n,n-k)$ for fixed $k$, proving that $DM(n,n-k) = O(n^k)$ and providing an exact formula for $DM(n,n-2)$.