New York Journal of Mathematics
Volume 2 (1996) 59-68

  

Doron Zeilberger

Proof of the Refined Alternating Sign Matrix Conjecture


Published: April 15, 1996
Keywords: enumeration, alternating sign matrices, square ice, Izergin-Korepin formula, orthogonal polynomials, q-analysis, q-Legendre 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!...(3n-2)!)/(n!(n+1)!...(2n-1)!).
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+r-2}\choose{n-1})({2n-1-r}\choose{n-1})]/({3n-2}\choose{n-1}).
Standing on the shoulders of A. G. Izergin, V. E. Korepin, and G. Kuperberg, and using in addition orthogonal polynomials and q-calculus, this stronger conjecture is proved.

Acknowledgements

Supported in part by the NSF.


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Author information

Department of Mathematics, Temple University, Philadelphia, PA19122, USA
Current Address: Department of Mathematics, Rutgers University, Hill Center-Busch Campus, Piscataway, NJ 08854-8019, USA

http://www.math.rutgers.edu/~zeilberg/