### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Drew Sills, Rutgers University, asills {at} math [dot] rutgers [dot] edu
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: The zero attractor of the partition polynomials

Speaker: William Goh, Drexel University

Date: September 29, 2005 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Many natural sequences of polynomials from combinatorics have interesting limiting sets of zeros in the complex plane. We will discuss one of the problems posed by Richard Stanley and Herb Wilf.

This is the sequence of partition polynomials $$F_{n}(x):=\sum_{k=1}^{n} p_k (n) x^k,$$ where $p_k (n)$ is the number of partitions of $n$ with exactly $k$ parts. In Stanley's plot for degree 200, the zeros cluster around the unit circle together with a sparse family inside the disc. Initially it was unclear what role such zeros play in the limit. Using techniques from analytic number theory and extensive numerical computation to degree 26,000, we found that the zeros approach the unit circle as well as three curves inside the disc given implicitly in terms of $f_{k}(z)=\Re(\sqrt{Li_{2}(z^{k})})/k$ where $Li_{2}$ is the dilogarithm. We will outline our attack on this problem in which we reduce the proof to a conjecture about domination among the functions $f_k$ which we verified numerically and in special cases.

(Joint work with Bob Boyer)