Species Count in Discrete, Continuous, and Infinite-Dimensional Composition Models

This work develops a unified theory of species count across three levels of composition models:  

(1) discrete stars-and-bars compositions,  

(2) continuous Dirichlet compositions, and  

(3) infinite-dimensional Poisson–Dirichlet compositions.  

We derive exact formulas for the discrete species count, introduce a natural threshold-based definition for the continuous case, and obtain the classical \(\theta \log(1/\varepsilon)\) law for the Poisson–Dirichlet distribution.  

Together, these results reveal a single structural mechanism underlying species accumulation phenomena in finite and infinite-dimensional composition structures.