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Alternating Sign Matrix Conjecture
The conjecture that the number of alternating sign matrices "bordered" by 
s 
is explicitly given by the formula
 |
This conjecture was proved by Doron Zeilberger in 1995 (Zeilberger 1996a). This proof enlisted the aid of an army of 88 referees together with extensive computer calculations.
A beautiful, shorter proof was given later that year by Kuperberg (Kuperberg 1996),
and the refined alternating
sign matrix conjecture was subsequently proved by Zeilberger (Zeilberger 1996b)
using Kuperberg's method together with techniques from 
-calculus and orthogonal
polynomials.
See also
Alternating Sign Matrix, Refined Alternating Sign Matrix ConjectureExplore with Wolfram|Alpha
References
Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999.Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637-646.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 413, 2003.Kuperberg, G. "Another Proof of the Alternating-Sign Matrix Conjecture." Internat. Math. Res. Notes, No. 3, 139-150, 1996.Zeilberger, D. "A Constant Term Identity Featuring the Ubiquitous (and Mysterious) Andrews-Mills-Robbins-Rumsey numbers 1, 2, 7, 42, 429, ...." J. Combin. Theory A 66, 17-27, 1994.Zeilberger, D. "Proof of the Alternating Sign Matrix Conjecture." Elec. J. Combin. 3, No. 2, R13, 1-84, 1996a. https://doi.org/10.37236/1271.Zeilberger, D. "Proof of the Refined Alternating Sign Matrix Conjecture." New York J. Math. 2, 59-68, 1996b.Referenced on Wolfram|Alpha
Alternating Sign Matrix ConjectureCite this as:
Weisstein, Eric W. "Alternating Sign Matrix Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlternatingSignMatrixConjecture.html