The Partition-Frequency Enumeration Matrix

TL;DR

This paper introduces the Partition-Frequency Enumeration matrix, a new calculus that unifies and extends results connecting number-theoretic functions with partition-like objects, leading to new recurrence relations and congruences.

Contribution

The paper develops an elementary calculus using the PFE matrix to unify and extend classical results on number-theoretic and partition functions, including new recurrence relations and congruences.

Findings

Unified framework for partition-like objects and number-theoretic functions.

Extended classical recurrence relations for divisor sums, Ramanujan's tau, and zeta values.

Embedded Ramanujan's congruences into infinite families.

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, Ewell, Ramanujan, Lehmer 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.