  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.
