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.