We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite number-theoretic matrix. We call this matrix the partition-frequency enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-theoretic functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan’s 𝜏$\tau$ function, sums of squares and triangular numbers, and for 𝜁(2𝑛)$\zeta (2n)$, where n is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer, and others. As one application, we embed Ramanujan’s famous congruences 𝑝(5𝑛+4)≡0(mod5)$p(5n+4)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}5)$ and 𝜏(5𝑛+5)≡0(mod5)$\tau (5n+5)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}5)$ into an infinite family of such congruences.