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

Derivatives and Integrals of Polynomials Associated with Integer Partitions