Equidistribution and partition polynomials

TL;DR

This paper uses equidistribution criteria to prove divisibility properties of various partition polynomials, providing new proofs of Ramanujan-type congruences and analyzing root distributions on the unit circle.

Contribution

It introduces new methods to establish divisibility and equidistribution results for partition polynomials, extending previous work and suggesting further applications.

Findings

Partition polynomials are divisible by cyclotomic polynomials.

Roots of partition polynomials are equidistributed on the unit circle.

New proofs of Ramanujan-type congruences are obtained.

Abstract

Using equidistribution criteria, we establish divisibility by cyclotomic polynomials of several partition polynomials of interest, including $spt$-crank, overpartition pairs, and $t$-core partitions. As corollaries, we obtain new proofs of various Ramanujan-type congruences for associated partition functions. Moreover, using results of Erd\"os and Tur\'an, we establish the equidistribution of roots of partition polynomials on the unit circle including those for the rank, crank, $spt$, and unimodal sequences. Our results complement earlier work on this topic by Stanley, Boyer-Goh, and others. We explain how our methods may be used to establish similar results for other partition polynomials of interest, and offer many related open questions and examples.