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Derivatives and Integrals of Polynomials Associated with Integer Partitions
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Madeline Locus Dawsey, Tyler Russell, and Dannie Urban

Department of Mathematics

University of Texas at Tyler

Tyler, TX 75799

USA

**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*_{λ}(*x*) that we call
the partition polynomial for the partition λ, with the aim
to learn new properties of partitions. We prove a recursive formula
for the derivatives of *f*_{λ}(*x*) involving Stirling numbers of the
second kind, show that the set of integrals from 0 to 1 of a normalized
version of *f*_{λ}(*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.

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(Concerned with sequences
A000041
A008277
A305078.)

Received August 2 2021; revised versions received March 9 2022; June 7 2022.
Published in *Journal of Integer Sequences*,
June 10 2022.

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