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.
Full version: pdf,
(Concerned with sequences
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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