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Enumerations of half-turn-symmetric alternating-sign matrices of odd order. (English) Zbl 1177.15041
Theor. Math. Phys. 148, No. 3, 1174-1198 (2006); translation from Teor. Mat. Fiz. 148, No. 3, 357-386 (2006).
Summary: Kuperberg showed that the partition function of the square-ice model related to half-turn-symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijective correspondence with half-turn-symmetric alternating-sign matrices of odd order. The partition function of this model is expressed via the above factors. We find the contributions to the partition function that correspond to the alternating-sign matrices having \(1\) or \(- 1\) as the central entry and establish the related enumerations.

MSC:
15B57 Hermitian, skew-Hermitian, and related matrices
05A15 Exact enumeration problems, generating functions
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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[1] W. H. Mills, D. P. Robbins, and H. Rumsey, Invent. Math., 66, 73–87 (1982); W. H. Mills, D. P. Robbins, and H. Rumsey, J. Combin. Theory Ser. A, 34, 340–359 (1983). · Zbl 0477.05011 · doi:10.1007/BF01404757
[2] G. Kuperberg, Internat. Math. Res. Notices, 3, 139–150 (1996); math.CO/9712207 (1997). · Zbl 0859.05027 · doi:10.1155/S1073792896000128
[3] G. Kuperberg, Ann. Math., 156, 835–866 (2002); math.CO/0008184 (2000). · Zbl 1010.05014 · doi:10.2307/3597283
[4] D. P. Robbins, ”Symmetry classes of alternating sign matrices,” math.CO/0008045 (2000).
[5] V. E. Korepin, Comm. Math. Phys., 86, 391–418 (1982). · Zbl 0531.60096 · doi:10.1007/BF01212176
[6] D. P. Robbins and H. Rumsey, Adv. Math., 62, 169–184 (1986); N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, J. Algebraic Combin., 1, 111–132, 219–234 (1992); math.CO/9201305 (1992). · Zbl 0611.15008 · doi:10.1016/0001-8708(86)90099-X
[7] Yu. G. Stroganov, Theor. Math. Phys., 146, 53–62 (2006); math-ph/0204042 (2002); A. V. Razumov and Yu. G. Stroganov, Theor. Math. Phys., 141, 1609–1630 (2004); math-ph/0312071 (2003). · Zbl 1177.82042 · doi:10.1007/s11232-006-0006-8
[8] Yu. G. Stroganov, ”Izergin-Korepin determinant reloaded,” math-ph/0409072 (1994).
[9] S. Okada, J. Algebraic Combin., 23, 43–69 (2006); math.CO/0408234 (2004). · Zbl 1088.05012 · doi:10.1007/s10801-006-6028-3
[10] G. Kuperberg, E-mail message to private ”domino” forum, 10 July 2004; for access to forum contact Jim Propp at proppmath.wisc.edu (2004).
[11] A. G. Izergin, Sov. Phys. Dokl., 32, No. 11, 878–879 (1987); V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method, Correlation Functions, and Algebraic Bethe Ansatz (2nd ed.), Cambridge Univ. Press, New York (1993).
[12] D. Zeilberger, Elec. J. Comb., 3(2), R13 (1996); math.CO/9407211 (1994).
[13] D. Zeilberger, New York J. Math., 2, 59–68 (1996); math.CO/9606224 (1996).
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