Determinant formulae for some tiling problems and application to fully packed loops
[Formules de déterminants pour quel\-ques problèmes de pavage et application aux modèles de boucles compactes]
Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2025-2050.

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.

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.

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
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)
@article{AIF_2005__55_6_2025_0,
     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},
     number = {6},
     year = {2005},
     doi = {10.5802/aif.2150},
     mrnumber = {2187944},
     zbl = {1075.05007},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2150/}
}
TY  - JOUR
AU  - Di Francesco, Philippe
AU  - Zinn-Justin, Paul
AU  - Zuber, Jean-Bernard
TI  - Determinant formulae for some tiling problems and application to fully packed loops
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 2025
EP  - 2050
VL  - 55
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2150/
DO  - 10.5802/aif.2150
LA  - en
ID  - AIF_2005__55_6_2025_0
ER  - 
%0 Journal Article
%A Di Francesco, Philippe
%A Zinn-Justin, Paul
%A Zuber, Jean-Bernard
%T Determinant formulae for some tiling problems and application to fully packed loops
%J Annales de l'Institut Fourier
%D 2005
%P 2025-2050
%V 55
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2150/
%R 10.5802/aif.2150
%G en
%F AIF_2005__55_6_2025_0
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, Tome 55 (2005) no. 6, pp. 2025-2050. doi : 10.5802/aif.2150. http://www.numdam.org/articles/10.5802/aif.2150/

[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=-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

Cité par Sources :