On the Zeros of Plane Partition Polynomials

Abstract

Let $PL(n)$ be the number of all plane partitions of $n$ while $pp_k(n)$ be the number of plane partitions of $n$ whose trace is exactly $k$. We study the zeros of polynomial versions $Q_n(x)$ of plane partitions where $Q_n(x) = \sum pp_k(n) x^k$. Based on the asymptotics we have developed for $Q_n(x)$ and computational evidence, we determine the limiting behavior of the zeros of $Q_n(x)$ as $n\to\infty$. The distribution of the zeros has a two-scale behavior which has order $n^{2/3}$ inside the unit disk while has order $n$ on the unit circle.