Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to $\omega(t)$

TL;DR

This paper explores the properties of integer partitions into fractions with fixed denominators and varying numerators, revealing symmetry, unimodality, and connections to number theory functions like $\omega(t)$ and Dirichlet series.

Contribution

It introduces new results on the symmetry and unimodality of partitions with odd numerators and establishes formulas linking partitions to the prime omega function extension.

Findings

Partitions with odd numerators exhibit symmetric patterns.

The generating function's nonzero terms are unimodal with a peak at h.

Derived formulas connect partitions to the prime omega function and Dirichlet series.

Abstract

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to $h$, then the nonzero terms of the generating function are unimodal, with the integer $h$ having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerators. We prove that there are $2^{\omega(t)}-2$ partitions of an integer $t$ into fractions with the first $x$ consecutive even integers for numerators and equal denominators of $y$, where $0<y<x<t$. We then use this to produce corollaries such as a Dirichlet series identity and an extension of the prime omega function to the complex plane, though this extension is not analytic everywhere.