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Colored vertex models and $k$-tilings of the Aztec diamond
Combinatorics · Submitted 12 Feb 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $t\rightarrow\infty$.
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The abstract, sentence by sentence
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 $t\rightarrow\infty$. We also present some lozenge $k$-tilings of the hexagon and compute the arctic curves of the tilings for $t=0$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$. | contribution 0.88 | 0.85 | 0.05 | asserted 0.92 |
| s1 | 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. | method 0.74 | 0.93 | 0.05 | asserted 0.99 |
| s2 | Employing the colored vertex models introduced in earlier work to study supersymmetric LLT polynomials, we compute the generating polynomials of the $k$-tilings. | method 0.89 | 0.92 | 0.18 | asserted 1.00 |
| s3 | 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 $t\rightarrow\infty$. | result 0.80 | 0.96 | 0.28 | asserted 1.00 |
| s4 | We also present some lozenge $k$-tilings of the hexagon and compute the arctic curves of the tilings for $t=0$. | contribution 0.74 | 0.94 | 0.07 | asserted 1.00 |
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The paper
Original 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 $t\rightarrow\infty$. We also present some lozenge $k$-tilings of the hexagon and compute the arctic curves of the tilings for $t=0$.
Human review
How the literature uses it
4 arXiv papers cite this in 13 sentences
Papers citing it · 7 sentences without a confident label
Builds on it 1 paper
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In this article we generalize the domino shuffling algorithm for tilings of the Aztec diamond to shuffling algorithm for interacting $k$-tilings studied in [ref] .
Shuffling algorithm for coupled tilings of the Aztec diamond · Conclusion · 1.00
Compares with it 1 paper
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One of the statistics they considered is based on the number of “coupled dominoes” (see [ref] ), which is analogues to our “domino pairs” (see Figure [ref]), although the two concepts are different.
Domino Tilings, Domino Shuffling, and the Nabla Operator · Introduction · 0.99
Cites it as background 1 paper
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We state some fundamental results from [ref] .
Shuffling algorithm for coupled tilings of the Aztec diamond · Introduction · 0.88
What it cites · 5 sentences without a confident label
Uses the method or tool of 2 papers
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By relating these tilings to the colored vertex models introduced in [ref] , we prove that The generating polynomial of the $k$-tilings of the Aztec diamond of rank $m$ is [equation] Here $t$ follows the number of interactions, and $x_i$ and $y_j$ follow the numbers of dominos of certain types as defined in Section [ref].
A Vertex Model for Supersymmetric LLT Polynomials · Introduction · 0.88
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We will use the vertex models introduced in [ref] , which can also be realized as degenerations of a vertex model introduced in [ref] .
Colored Fermionic Vertex Models and Symmetric Functions · Introduction · 0.96
Cites as background 2 papers
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These vertex models were originally defined and studied in [ref] , although they can be realized as degenerations of vertex models introduced in [ref] (see [ref] ).
Colored Fermionic Vertex Models and Symmetric Functions · Defining the vertex models · 0.90
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These vertex models were originally defined and studied in [ref] , although they can be realized as degenerations of vertex models introduced in [ref] (see [ref] ).
A Vertex Model for Supersymmetric LLT Polynomials · Defining the vertex models · 0.90
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