Journal of Integer Sequences, Vol. 25 (2022), Article 22.5.1

Derivatives and Integrals of Polynomials Associated with Integer Partitions

Madeline Locus Dawsey, Tyler Russell, and Dannie Urban
Department of Mathematics
University of Texas at Tyler
Tyler, TX 75799


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 fλ(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 fλ(x) involving Stirling numbers of the second kind, show that the set of integrals from 0 to 1 of a normalized version of fλ(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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