Treatise on Restricted Partitions

Abstract

This paper introduces Sivanendran’s Restricted Partition Function (SRPF), a novel framework for constructing and enumerating restricted partitions of natural numbers. Unlike contemporary approaches, SRPF relies on simple floor functions and integer division, making it computationally efficient even for large numbers. Iteration variables directly encode part multiplicities, enabling instant construction of partitions rather than mere enumeration—a capability absent in existing algorithms. Complementing SRPF, the Factor Partition Set provides an elegant lens to deduce primality, squareness (or powers in general), and factorisation properties from partition symmetries, acting as an indexed, compressed information system. Together, these methods establish a constructive, algorithmic paradigm for partition theory that is both faster and richer in structural insight than classical analytic techniques.