Alternating sign matrix: Difference between revisions

Revision as of 11:58, 30 January 2019

Revision as of 16:43, 4 March 2019

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</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 conjecture 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>.

==Razumov–Stroganov conjecture==

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==

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* Mills, William H., 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., [https://arxiv.org/abs/cond-mat/0108103 O(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices], ''Theor. Math. Phys.'', '''142''' (2005), 237–243.

* Robbins, David P., The story of <math>1, 2, 7, 42, 429, 7436, \dots</math>, ''The Mathematical Intelligencer'', 13 (2), 12–19 (1991).

* 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.