Determinant formulae for some tiling problems and application to fully packed loops

Di Francesco, Philippe; Zinn-Justin, Paul; Zuber, Jean-Bernard

doi:10.5802/aif.2150

Di Francesco, Philippe ¹; Zinn-Justin, Paul; Zuber, Jean-Bernard

¹ CEA-Saclay, service de physique théorique de Saclay, CEA/DSM/SPhT, URA 2306 du CNRS, 91191 Gif sur Yvette Cedex (France), Independent University, LIFR-MIIP, 119002, Bolshoy Vlasyevskiy Pereulok 11, Moscow (Russie), Université Paris-Sud, laboratoire de physique théorique et modèles statistiques, UMR 8626 du CNRS, bâtiment 100, 91405 Orsay Cedex (France), Université Paris 6, LPTHE, Tour 24, 75231 Paris Cedex 05 (France)

Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 2025-2050.

Abstract
Résumé

We present a number of determinant formulae for the number of tilings of various domains in relation with Alternating Sign Matrix and Fully Packed Loop enumeration.

Quelques formules de déterminants sont données pour le dénombrement des pavages dans différents domaines, en relation avec les énumérations de matrices à signes alternés et de boucles compactes.

MR Zbl

DOI: 10.5802/aif.2150

Classification: 05A19, 52C20, 82B20
Keywords: Tilings, alternating sign matrices, fully packed loops
Mot clés : pavages, matrices à signes alternés, boucles compactes

Author's affiliations:

Di Francesco, Philippe ¹; Zinn-Justin, Paul ; Zuber, Jean-Bernard

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     author = {Di Francesco, Philippe and Zinn-Justin, Paul and Zuber, Jean-Bernard},
     title = {Determinant formulae for some tiling problems and application to fully packed loops},
     journal = {Annales de l'Institut Fourier},
     pages = {2025--2050},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
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     doi = {10.5802/aif.2150},
     mrnumber = {2187944},
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     language = {en},
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Di Francesco, Philippe; Zinn-Justin, Paul; Zuber, Jean-Bernard. Determinant formulae for some tiling problems and application to fully packed loops. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 2025-2050. doi : 10.5802/aif.2150. http://archive.numdam.org/articles/10.5802/aif.2150/

References
Cited by

[1] M. Adler; P. van Moerbeke Virasoro action on Schur function expansions, skew Young tableaux and random walks, Comm. Pure Appl. Math., Volume 58 (2005), pp. 362-408 | DOI | MR | Zbl

[2] M.T. Batchelor; J. de Gier; B. Nienhuis The quantum symmetric XXZ chain at $Δ = - \frac{1}{2}$ , alternating sign matrices and plane partitions, J. Phys. A, Volume 34 (2001), p. L265-L270 | DOI | MR | Zbl

[3] D. Bressoud Proofs and confirmations. The story of the alternating sign matrix conjecture, Cambridge University Press, 1999 | MR | Zbl

[4] F. Caselli; C. Krattenthaler Proof of two conjectures of Zuber on fully packed loop configurations, J. Combin. Theory Ser. A, Volume 108 (2004), pp. 123-146 | DOI | MR | Zbl

[5] M. Ciucu; C. Krattenthaler Enumeration of lozenge tilings of hexagons with cut-off corners, J. Combin. Theory Ser. A, Volume 100 (2002), pp. 201-231 | DOI | MR | Zbl

[6] M. Ciucu; C. Krattenthaler; T. Eisenkölbl; D. Zare Enumeration of lozenge tilings of hexagons with a central triangular hole, J. Combin. Theory Ser. A, Volume 95 (2001), pp. 251-334 | DOI | MR | Zbl

[7] P. Di Francesco; P. Zinn-Justin; J.-B. Zuber A bijection between classes of fully packed loops and plane partitions, Electron. J. Combin., Volume 11 (2004) no. 1 | MR | Zbl

[8] P. Di Francesco; J.-B. Zuber On FPL configurations with four sets of nested arches, JSTAT (2004) | Zbl

[9] J. de Gier Loops, matchings and alternating-sign matrices (math.CO/0211285), http://arxiv.org/abs/math.CO/0211285 | Zbl

[10] J. Grassberger; A. King; P. Tirao On the homology of free 2-step nilpotent Lie algebras, J. Algebra, Volume 254 (2002), pp. 213-225 | DOI | MR | Zbl

[11] C. Krattenthaler Advanced determinant calculus, Séminaire Lotharingien Combin., 42, 1999 | MR | Zbl

[12] B. Lindström On the vector representations of induced matroids, Bull. London Math. Soc., Volume 5 (1973), pp. 85-90 | DOI | MR | Zbl

[13] S. Mitra; B. Nienhuis Osculating random walks on cylinders, pp. 259-264 in Discrete random walks, DRW'03, Discrete Mathematics and Computer Science Proceedings AC, 2003 | MR | Zbl

[14] S. Mitra; B. Nienhuis; J. de Gier; M.T. Batchelor Exact expressions for correlations in the ground state of the dense $O (1)$ loop model, JSTAT (2004) | Zbl

[15] On-Line Encyclopedia of Integer Sequences (, http://www.research.att.com/~njas/sequences/Seis.html)

[16] P.A. Pearce; V. Rittenberg; J. de Gier Critical $Q = 1$ Potts model and temperley-Lieb stochastic processes, Teor. Mat. Fiz., Volume 142 (2005), pp. 284-292

[17] A.V. Razumov; Yu.G. Stroganov Combinatorial nature of ground state vector of O(1) loop model, Theor. Math. Phys., Volume 138 (2004), pp. 333-337 | DOI | MR | Zbl

[18] B. Wieland A large dihedral symmetry of the set of alternating-sign matrices, Electron. J. Combin., Volume 7 (2000) | MR | Zbl

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