 

Doron Zeilberger
Proof of the Refined Alternating Sign Matrix Conjecture


Published: 
April 15, 1996 
Keywords: 
enumeration, alternating sign matrices, square ice, IzerginKorepin formula, orthogonal polynomials, qanalysis, qLegendre polynomials, exactly solvable models 
Subject: 
Primary: 05A; Secondary: 33 


Abstract
Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that
the number of alternating sign matrices of order $n$ equals
A(n):=(1!4!7!...(3n2)!)/(n!(n+1)!...(2n1)!).
Mills, Robbins, and Rumsey also made the stronger conjecture that
the number of such matrices
whose (unique) '1' of the first row is at the rth column
equals
A(n)[({n+r2}\choose{n1})({2n1r}\choose{n1})]/({3n2}\choose{n1}).
Standing on the shoulders of A. G. Izergin, V. E. Korepin, and G. Kuperberg,
and using in addition orthogonal polynomials and qcalculus,
this stronger conjecture is proved.


Acknowledgements
Supported in part by the NSF.


Links


Author information
Department of Mathematics, Temple University,
Philadelphia, PA19122, USA
Current Address: Department of Mathematics, Rutgers University, Hill CenterBusch Campus, Piscataway, NJ 088548019, USA
http://www.math.rutgers.edu/~zeilberg/

