A new simple proof of the Aztec diamond theorem

Daniel Grieser; Manuel Fendler

RECORD

A new simple proof of the Aztec diamond theorem

Manuel Fendler, Daniel Grieser

Combinatorics · Submitted 21 Oct 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We give a new simple proof of this theorem.

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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,517 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.04 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.62 · compares with prior results 0.68 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Aztec diamond of order $n$ is the union of lattice squares in the plane intersecting the square $|x|+|y|<n$. The Aztec diamond theorem states that the number of domino tilings of this shape is $2^{n(n+1)/2}$. It was first proved by Elkies, Kuperberg, Larsen and Propp in 1992. We give a new simple proof of this theorem.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Aztec diamond of order $n$ is the union of lattice squares in the plane intersecting the square $\|x\|+\|y\|<n$.	background 1.00	0.08	0.02	asserted 1.00
s1	The Aztec diamond theorem states that the number of domino tilings of this shape is $2^{n(n+1)/2}$.	background 0.95	0.06	0.28	asserted 1.00
s2	It was first proved by Elkies, Kuperberg, Larsen and Propp in 1992.	background 1.00	0.03	0.02	asserted 1.00
s3	We give a new simple proof of this theorem.	contribution 1.00	0.95	0.71	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

The Aztec diamond of order $n$ is the union of lattice squares in the plane intersecting the square $|x|+|y|<n$. The Aztec diamond theorem states that the number of domino tilings of this shape is $2^{n(n+1)/2}$. It was first proved by Elkies, Kuperberg, Larsen and Propp in 1992. We give a new simple proof of this theorem.

Human review

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How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given in terms of Hankel determinants of the large and small Schr\"oder numbers.

A Simple Proof of the Aztec Diamond Theorem · 0.99

Papers citing it

Cites it as background 1 paper

See [ref] for the four original proofs; and e.g. [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] for further proofs.

A note on a 2-enumeration of antisymmetric monotone triangles · Introduction · 0.98

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