Twenty Vertex model and domino tilings of the Aztec triangle
The electronic journal of combinatorics, Tome 28 (2021) no. 4
We show that the number of configurations of the 20 Vertex model on certain domains with domain wall type boundary conditions is equal to the number of domino tilings of Aztec-like triangles, proving a conjecture of the author and Guitter. The result is based on the integrability of the 20 Vertex model and uses a connection to the U-turn boundary 6 Vertex model to re-express the number of 20 Vertex configurations as a simple determinant, which is then related to a Lindström-Gessel-Viennot determinant for the domino tiling problem. The common number of configurations is conjectured to be $2^{n(n-1)/2}\prod_{j=0}^{n-1}\frac{(4j+2)!}{(n+2j+1)!}=1, 4, 60, 3328, 678912...$ The enumeration result is extended to include refinements of both numbers.
DOI :
10.37236/10227
Classification :
05A15, 05B45, 82B20, 82B27
Mots-clés : triangular ice model combinatorics, 20V-domino tiling correspondence
Mots-clés : triangular ice model combinatorics, 20V-domino tiling correspondence
Affiliations des auteurs :
Philippe Di Francesco 1
@article{10_37236_10227,
author = {Philippe Di Francesco},
title = {Twenty {Vertex} model and domino tilings of the {Aztec} triangle},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/10227},
zbl = {1486.05014},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10227/}
}
Philippe Di Francesco. Twenty Vertex model and domino tilings of the Aztec triangle. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/10227
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