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ABSTRACT
We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular 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-type 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 \tau function, sums of squares and triangular numbers, and for \zeta(2n), where n is a positive integer. These include classical results due to Euler, Ramanujan, and others. As one application, we embed Ramanujan's famous congruences p(5n+4)\equiv 0\; (mod 5) and \tau(5n+5)\equiv 0\; (mod 5) into an infinite family of such congruences.
This is joint work with Hartosh Singh Bal.
