Selective Multiset Combination: A Combinatorial Foundation for Finite EPPFs and Their Continuous Limits

This work introduces Selective Multiset Combination, a new combinatorial construction that decomposes the multiset coefficient \(\binom{n+k-1}{k}\) by the number of active categories. The resulting identity is both a natural occupancy-based decomposition and a special case of Vandermonde’s convolution. This structure induces a finite exchangeable partition probability function (EPPF), whose continuous limits recover the Dirichlet distribution (continuous stars-and-bars) and, under symmetric Dirichlet scaling, the Poisson–Dirichlet distribution. The paper provides a unified finite-to-continuous bridge connecting combinatorics, occupancy distributions, Dirichlet laws, and exchangeable partition structures.