### 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: Disturbing the Dyson Conjecture, in a Generally Good Way

Speaker: Drew Sills, Rutgers University

Date: October 27, 2005 5:00pm

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

Abstract:

Let F(n) := F_n( x_1, . . . , x_n ; a_1, . . . , a_n ):= \prod_{1\leq i\neq j \leq n) ( 1 - x_i / x_j )^{a_j}. In 1962, Freeman Dyson conjectured that the constant term in the expansion of F(n) is the multinomial coefficient (a_1 + a_2 + . . . + a_n)!/(a_1! a_2! . . . a_n!). In 1975, George Andrews extended Dyson's conjecture to a q-analog. A particularly elegant proof of Dyson's conjecture was given by I. J. Good in 1970. Good's proof does not extend to the q-analog, however, and the q-Dyson conjecture was not settled until 1985 when Zeilberger and Bressoud proved it combinatorially.

Last March in the Experimental Mathematics Seminar, I demonstrated how a Maple package that I developed with Professor Zeilberger could be used to automatically conjecture and prove closed form expressions for coefficients in the expansion of F(n) besides the constant term, for fixed n. The automated proofs are based on a generalization of Good's proof. In this lecture, I will discuss more recent work, where the "disturbed Dyson conjectures" and their proofs are extended to symbolic n, and corresponding q-analogs are conjectured.