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</math>
==Alternating sign matrix conjecturetheorem==
The ''alternating sign matrix conjecturetheorem'' states that the number of <math>n\times n</math> alternating sign matrices is
:<math>
\prod_{k=0}^{n-1}\frac{(3k+1)!}{(n+k)!} = \frac{1!\, 4! \,7! \cdots (3n-2)!}{n!\, (n+1)! \cdots (2n-1)!}.
</math>
The first few terms in this sequence for ''n'' = 0, 1, 2, 3, … are
:[[1 (number)|1]], 1, [[2 (number)|2]], [[7 (number)|7]], [[42 (number)|42]], 429, 7436, 218348, … {{OEIS|id=A005130}}.
This conjecturetheorem was first proved by [[Doron Zeilberger]] in 1992.<ref>Zeilberger, Doron, [http://www.combinatorics.org/Volume_3/Abstracts/v3i2r13.html "Proof of the alternating sign matrix conjecture"], ''[http://www.combinatorics.org/ Electronic Journal of Combinatorics]'' 3 (1996), R13.</ref> In 1995, [[Greg Kuperberg]] gave a short proof<ref>[[Greg Kuperberg|Kuperberg, Greg]], [http://front.math.ucdavis.edu/math.CO/9712207 "Another proof of the alternating sign matrix conjecture"], ''International Mathematics Research Notes'' (1996), 139-150.</ref> based on the [[Yang–Baxter equation]] for the six-vertex model with ___domain-wall boundary conditions, that uses a determinant calculation due to Anatoli Izergin.<ref>"Determinant formula for the six-vertex model", A. G. Izergin et al. 1992 ''J. Phys. A'': Math. Gen. 25 4315.</ref> A third proof was given by [[Ilse Fischer]] using what is called the ''operator method''.<ref>{{Cite journal|last=Fischer|first=Ilse|title=A new proof of the refined alternating sign matrix theorem|journal=Journal of Combinatorial Theory, Series A|year=2005|volume=114|issue=2|pages=253–264|doi=10.1016/j.jcta.2006.04.004|arxiv=math/0507270|bibcode=2005math......7270F}}</ref>
==Razumov–Stroganov conjecture==
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