View source for Alternating sign matrix

{{distinguish|Alternant matrix}}
{{Image frame|width=340|align=right|caption=The seven alternating sign matrices of size 3
|content=<math>\begin{matrix}
 \begin{bmatrix}
  1 & 0 & 0\\
  0 & 1 & 0\\
  0 & 0 & 1
 \end{bmatrix}
 \qquad
 \begin{bmatrix}
  1 & 0 & 0\\
  0 & 0 & 1\\
  0 & 1 & 0
 \end{bmatrix}
\\
 \begin{bmatrix}
  0 & 1 & 0\\
  1 & 0 & 0\\
  0 & 0 & 1
 \end{bmatrix}
 \qquad
 \begin{bmatrix}
  0 &  1 & 0\\
  1 & -1 & 1\\
  0 &  1 & 0
 \end{bmatrix}
 \qquad
 \begin{bmatrix}
  0 & 1 & 0\\
  0 & 0 & 1\\
  1 & 0 & 0
 \end{bmatrix}
\\
 \begin{bmatrix}
  0 & 0 & 1\\
  1 & 0 & 0\\
  0 & 1 & 0
 \end{bmatrix}
 \qquad
 \begin{bmatrix}
  0 & 0 & 1\\
  0 & 1 & 0\\
  1 & 0 & 0
 \end{bmatrix}
\end{matrix}</math>}}
In [[mathematics]], an '''alternating sign matrix''' is a [[square matrix]] of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize [[Permutation matrix|permutation matrices]] and arise naturally when using [[Dodgson condensation]] to compute a determinant.<ref>{{citation
 | last = Hone | first = Andrew N. W.
 | doi = 10.1098/rsta.2006.1887
 | issue = 1849
 | journal = Philosophical Transactions of the Royal Society of London
 | mr = 2317901
 | pages = 3183–3198
 | title = Dodgson condensation, alternating signs and square ice
 | volume = 364
 | year = 2006}}</ref> They are also closely related to the [[six-vertex model]] with domain wall boundary conditions from [[statistical mechanics]]. They were first defined by William Mills, [[David P. Robbins|David Robbins]], and Howard Rumsey in the former context.

==Examples==

A [[permutation matrix]] is an alternating sign matrix, and an alternating sign matrix is a permutation matrix if and only if no entry equals {{math|−1}}.

An example of an alternating sign matrix that is not a permutation matrix is
[[File:Matrice signes alternants 4x4.svg|thumbnail|Puzzle picture]]
:<math>
\begin{bmatrix} 
0&0&1&0\\
1&0&0&0\\
0&1&-1&1\\
0&0&1&0
\end{bmatrix}.
</math>

==Alternating sign matrix theorem==
The ''alternating sign matrix theorem'' 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, 1, 2, 7, 42, 429, 7436, 218348, … {{OEIS|id=A005130}}.

This theorem 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://arxiv.org/abs/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> In 2005, 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 problem==

In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model (FPL) and ASMs.<ref>Razumov, A.V., Stroganov Yu.G., [https://arxiv.org/abs/cond-mat/0012141 Spin chains and combinatorics], ''Journal of Physics A'', '''34''' (2001), 3185-3190.</ref>
This conjecture was proved in 2010 by Cantini and Sportiello.<ref>L. Cantini and A. Sportiello, [https://arxiv.org/abs/1003.3376 Proof of the Razumov-Stroganov conjecture]''Journal of Combinatorial Theory, Series A'', '''118 (5)''', (2011) 1549–1574,</ref>

==References==
{{reflist}}

==Further reading==
* [[David Bressoud|Bressoud, David M.]], ''Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture'', MAA Spectrum, Mathematical Associations of America, Washington, D.C., 1999.{{ISBN|978-0521666466}}
* [[David Bressoud|Bressoud, David M.]] and Propp, James, [https://www.ams.org/notices/199906/fea-bressoud.pdf How the alternating sign matrix conjecture was solved], ''Notices of the American Mathematical Society'', 46 (1999), 637–646.
* Mills, William H., [[David P. Robbins|Robbins, David P.]], and Rumsey, Howard Jr., Proof of the Macdonald conjecture, ''Inventiones Mathematicae'', 66 (1982), 73–87.
* Mills, William H., [[David P. Robbins|Robbins, David P.]], and Rumsey, Howard Jr., Alternating sign matrices and descending plane partitions, ''Journal of Combinatorial Theory, Series A'', 34 (1983), 340–359.
* Propp, James, [https://arxiv.org/abs/math/0208125v1 The many faces of alternating-sign matrices], ''Discrete Mathematics and Theoretical Computer Science'', Special issue on ''Discrete Models: Combinatorics, Computation, and Geometry'' (July 2001).
* Razumov, A. V., Stroganov Yu. G., [https://arxiv.org/abs/math/0104216 Combinatorial nature of ground state vector of O(1) loop model], ''Theor. Math. Phys.'', '''138''' (2004), 333–337.
* Razumov, A. V., Stroganov Yu. G., O(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices], ''Theor. Math. Phys.'', '''142''' (2005), 237–243, {{arxiv|cond-mat/0108103}}
* [[David P. Robbins|Robbins, David P.]], The story of <math>1, 2, 7, 42, 429, 7436, \dots</math>, ''The Mathematical Intelligencer'', 13 (2), 12–19 (1991), {{doi|10.1007/BF03024081}}.
* [[Doron Zeilberger|Zeilberger, Doron]], [http://nyjm.albany.edu:8000/j/1996/2-4.pdf Proof of the refined alternating sign matrix conjecture], ''New York Journal of Mathematics'' 2 (1996), 59–68.

==External links==
* [http://mathworld.wolfram.com/AlternatingSignMatrix.html Alternating sign matrix] entry in [[MathWorld]]
* [http://www.findstat.org/AlternatingSignMatrices Alternating sign matrices] entry in the [http://www.findstat.org/ FindStat] database

{{Matrix classes}}

[[Category:Matrices]]
[[Category:Enumerative combinatorics]]