Disturbing the Dyson Conjecture (in a GOOD Way)

Andrew V. Sills; Doron Zeilberger

Disturbing the Dyson Conjecture (in a GOOD Way)

Andrew V. Sills, Doron Zeilberger

Mathematical Sciences

Rutgers, The State University of New Jersey

Research output: Contribution to conference › Presentation

Abstract

Joint work with Doron Zeilberger. In 1962, Freeman Dyson conjectured that the constant term in the Laurent polynomial ∏ 1≤i≠j≤n (1-x i / x j ) aj (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 Mathematica and Maple packages "GoodDyson") that automatically conjecture and then supply 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.

Original language	American English
State	Published - Mar 10 2005
Event	Rutgers Experimental Mathematics Seminar - New Brunswick, NJ Duration: Mar 10 2005 → …

Conference

Conference	Rutgers Experimental Mathematics Seminar
Period	03/10/05 → …

Keywords

Dyson Conjecture

DC Disciplines

Mathematics

Cite this

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title = "Disturbing the Dyson Conjecture (in a GOOD Way)",

abstract = " Joint work with Doron Zeilberger. In 1962, Freeman Dyson conjectured that the constant term in the Laurent polynomial ∏ 1≤i≠j≤n (1-x i / x j ) aj (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 Mathematica and Maple packages {"}GoodDyson{"}) that automatically conjecture and then supply 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. ",

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note = "Any physical experience provides only a small number of results, whereas a large number of parameters may vary. What is the {"}value{"} of the information obtained this way, if, for instance, we have 300 measures, where the whole experiment may involve 50 parameters, thus leading to possible states, if each parameter can take 10 values?; Rutgers Experimental Mathematics Seminar ; Conference date: 10-03-2005",

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AU - Sills, Andrew V.

AU - Zeilberger, Doron

N1 - Any physical experience provides only a small number of results, whereas a large number of parameters may vary. What is the "value" of the information obtained this way, if, for instance, we have 300 measures, where the whole experiment may involve 50 parameters, thus leading to possible states, if each parameter can take 10 values?

PY - 2005/3/10

Y1 - 2005/3/10

N2 - Joint work with Doron Zeilberger. In 1962, Freeman Dyson conjectured that the constant term in the Laurent polynomial ∏ 1≤i≠j≤n (1-x i / x j ) aj (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 Mathematica and Maple packages "GoodDyson") that automatically conjecture and then supply 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.

AB - Joint work with Doron Zeilberger. In 1962, Freeman Dyson conjectured that the constant term in the Laurent polynomial ∏ 1≤i≠j≤n (1-x i / x j ) aj (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 Mathematica and Maple packages "GoodDyson") that automatically conjecture and then supply 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.

KW - Dyson Conjecture

UR - http://sites.math.rutgers.edu/~bte14/expmath/archive05.html

M3 - Presentation

T2 - Rutgers Experimental Mathematics Seminar

Y2 - 10 March 2005

ER -

Disturbing the Dyson Conjecture (in a GOOD Way)

Abstract

Conference

Keywords

DC Disciplines

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