Derivatives and Integrals of Polynomials Associated with Integer Partitions

TL;DR

This paper introduces a new polynomial associated with integer partitions, explores its derivatives and integrals, and investigates their properties to deepen understanding of partition theory.

Contribution

It develops recursive formulas for derivatives of the partition polynomial, studies the density of its integrals, and proposes conjectures linking integrals to partition length.

Findings

Derivatives of the partition polynomial involve Stirling numbers of the second kind.

Normalized integrals of the polynomial are dense in [0, 1/2].

Open questions and conjectures relate integrals to partition length.

Abstract

Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial $f_\lambda(x)$ that we call the partition polynomial for the partition $\lambda$, with the aim to learn new properties of partitions. We prove a recursive formula for the derivatives of $f_\lambda(x)$ involving Stirling numbers of the second kind, show that the set of integrals from 0 to 1 of a normalized version of $f_\lambda(x)$ is dense in $[0,1/2]$, pose a few open questions, and formulate a conjecture relating the integral to the length of the partition. We also provide specific examples throughout to support our speculation that an in-depth analysis of partition polynomials could further strengthen our understanding of partitions.