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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>{{
| 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| bibcode = 2006RSPTA.364.3183H
}}</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==
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==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.
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{{Matrix classes}}
[[Category:Matrices (mathematics)]]
[[Category:Enumerative combinatorics]]
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