A Constructive Pipeline for the Decomposition of Cartesian Powers into Symmetric Configuration Classes

We develop a constructive decomposition of the Cartesian power A^r into symmetric equivalence classes determined by symbol multiplicities. The decomposition partitions A^r into disjoint configuration classes and provides an explicit reconstruction of the entire Cartesian power from these classes. Within this framework, integer partitions, Stirling numbers of the second kind, Bell numbers, and multinomial coefficients arise naturally as successive structural layers of the decomposition. These classical combinatorial quantities are thereby organized within a common set-theoretic construction linking multiplicity profiles, partition structures, and multiset realizations. The framework is formulated through a sequence of set-theoretic operations that generate multiplicity profiles, symbol assignments, multiplicity distributions, and multiset permutations. Together these constructions provide a unified perspective on the combinatorial structure underlying Cartesian powers.