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Robert P. Boyer

Publications and source records attributed to Robert P. Boyer.

4 recordsLinked to original sources
arXiv2021

Zero Attractors of Partition Polynomials

A partition polynomial is a refinement of the partition number p(n) whose coefficients count some special partition statistic. Just as partition numbers have useful asymptotics so do partition polynomials. In fact, their asymptotics determine the limiting behavior of their zeros which form a network of curves inside the unit disk. An important new feature in their study requires a detailed analysis of. the \root dilogarithm" given as the real part of the square root of the usual dilogarithm.

math.CO↗
arXiv2013

Stabilization of coefficients for partition polynomials

We find that a wide variety of families of partition statistics stabilize in a fashion similar to $p_k(n)$, the number of partitions of n with k parts, which satisfies $p_k(n) = p_{k+1}(n + 1), k \geq n/2$. We bound the regions of stabilization, discuss variants on the phenomenon, and give the limiting sequence in many cases as the coefficients of a single-variable generating function. Examples include many statistics that have an Euler product form, partitions with prescribed subsums, and plane overpartitions.

math.CO↗
arXiv2007

Partition Polynomials: Asymptotics and Zeros

Let $F_n(x)$ be the partition polynomial $\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of $F_n(x)$ have two scales of orders $n$ and $\sqrt{n}$ and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm.

math.CO↗