Twenty Vertex model and domino tilings of the Aztec triangle

TL;DR

This paper proves a conjecture linking the number of configurations in the 20 Vertex model with domino tilings of Aztec-like triangles, using integrability and determinant techniques.

Contribution

It establishes a precise enumeration equivalence between the 20 Vertex model configurations and domino tilings, extending previous conjectures with determinant formulas.

Findings

Number of 20 Vertex configurations equals domino tilings of Aztec-like triangles.

Derived a determinant formula relating the two combinatorial objects.

Extended enumeration to include refined counts.

Abstract

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 from [P. Di Francesco and E. Guitter, Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings, Elec. Jour. of Combinatorics 27 (2020), no. 2, P2.13]. 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 Lindstr\"om-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.