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The Partition-Frequency Enumeration Matrix

Hartosh Singh Bal, Gaurav Bhatnagar

TL;DR

The paper introduces the Partition-Frequency Enumeration (PFE) matrix as a unifying calculus that links arithmetic functions to partition-type objects via a simple, elementary matrix framework. By developing a general theory for constructing PFE matrices from arbitrary generating functions (including Weierstrass products) and exploring multiple product- and power-form scenarios, the authors derive classical recurrences (e.g., Euler, Ramanujan, zeta-related) and embed Ramanujan-type congruences into infinite families. The work provides a versatile toolkit for deriving recurrences, congruences, and product identities across number theory and combinatorics, with concrete applications to roots of generating functions, powers of generating functions, and infinite congruence families, as well as connections to theta, eta, and Jacobi identities. These results open avenues for systematic exploration of arithmetic-partition correspondences and for generating new identities via the PFE calculus.

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 $τ$ function, sums of squares and triangular numbers, and for $ζ(2n)$, 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 $τ(5n+5)\equiv 0$ (mod $5)$ into an infinite family of such congruences.

The Partition-Frequency Enumeration Matrix

TL;DR

The paper introduces the Partition-Frequency Enumeration (PFE) matrix as a unifying calculus that links arithmetic functions to partition-type objects via a simple, elementary matrix framework. By developing a general theory for constructing PFE matrices from arbitrary generating functions (including Weierstrass products) and exploring multiple product- and power-form scenarios, the authors derive classical recurrences (e.g., Euler, Ramanujan, zeta-related) and embed Ramanujan-type congruences into infinite families. The work provides a versatile toolkit for deriving recurrences, congruences, and product identities across number theory and combinatorics, with concrete applications to roots of generating functions, powers of generating functions, and infinite congruence families, as well as connections to theta, eta, and Jacobi identities. These results open avenues for systematic exploration of arithmetic-partition correspondences and for generating new identities via the PFE calculus.

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 function, sums of squares and triangular numbers, and for , where 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 (mod and (mod into an infinite family of such congruences.

Paper Structure

This paper contains 8 sections, 12 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Enumeration of Partitions
  3. Connection with arithmetic functions
  4. A calculus for obtaining PFE Matrices
  5. Application 1: Roots of generating functions
  6. Application 2: Powers of generating functions
  7. Application 3: Infinite families of congruences
  8. Concluding Remarks

Key Result

Lemma 2.1

Let ${ \boldsymbol A}$ be an infinite matrix, and $A_n$ be the sub-matrix formed by taking the first $n$ columns of ${ \boldsymbol A}$. Suppose that $A_n$ has a finite number (say, $m_n$) of non-zero rows. Let $U_k$ and $V_k$ be some sequences of complex numbers, with $V_k\neq 0$ for all $k$. Let $P Then, given the initial condition $P(0)$, the above equations determine $P(n)$, and $F_k(n)$, for $

Theorems & Definitions (46)

  • Lemma 2.1: An enumeration lemma
  • proof
  • Example 2.1: Enumeration of partitions
  • Theorem 2.2
  • Remark
  • proof
  • Theorem 2.3
  • proof
  • Remark
  • Example 2.2: Overpartitions
  • ...and 36 more