Colored vertex models and k-tilings of the Aztec diamond

2022/02/12 by Sylvie Corteel, Corteel, Sylvie, Andrew Gitlin +3 · 2 citations
Mathematics · #05A15 #05E05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2202.06020

openalex publication_date 2022/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study k-tilings (k-tuples of domino tilings) of the Aztec diamond of rank m. We assign a weight to each k-tiling, depending on the number of dominos of certain types and the number of "interactions" between the tilings. Employing the colored vertex models introduced in earlier work to study supersymmetric LLT polynomials, we compute the generating polynomials of the k-tilings. We then prove some combinatorial results about k-tilings, including a bijection between k-tilings with no interactions and 1-tilings, and we compute the arctic curves of the tilings for t=0 and t→∞. We also present some lozenge k-tilings of the hexagon and compute the arctic curves of the tilings for t=0.

Cited by

• Limit shapes for Domain-Wall (colored) vertex models
• Colored Vertex Models and Interacting Reverse Plane Partitions

Related

• Domino tilings of the Aztec diamond with doubly periodic weightings
• A generalization of Aztec diamond theorem, part I
• A bijection proving the Aztec diamond theorem by combing lattice paths
• Generating function of the tilings of Aztec rectangle with holes
• A Generalization of Aztec Diamond Theorem, Part II