### 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 "GOOD" way)

Speaker: Drew Sills, Rutgers University

Date: March 10, 2005 4:30-5:30pm

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

Abstract:

In 1962, Freeman Dyson conjectured that the constant term in the Laurent polynomial $$\prod_{1\leq i\neq j\leq n} (1 - x_i / x_j )^{a_j}$$ (let us call this the "Dyson product") is the multinomial coefficient (a_1 + a_2 + . . . + a_n )!/ (a_1! a_2! . . . a_n!). Dyson's conjecture was first proved independently by Gunson and Wilson. The most compact and elegant proof, however, was supplied by I.J. Good in 1970. We present a case study in experimental yet rigorous mathematics by describing an algorithm (which we have fully implemented in the Maple package "GoodDyson") that automatically conjectures and then supplies proofs (inspired by Good's proof) of closed form expressions for extensions of Dyson's conjecture to coefficients beside the constant term in the Dyson product.

(Joint work with Doron Zeilberger.)