Convert tilings to matchings
Represent domino tilings as perfect matchings and write the weighted count as a signed determinant.
← All researchMF-PRISM-MATH-2026-14 / interactive guide
THE IDEA, MADE VISIBLE
A whole space of domino arrangements.
Explore an exactly enumerable Aztec diamond—a baseline for understanding more demanding tiling regions.
Candidate resolutionCatalan Kernels for Augmented Aztec Rectangles
Specify the region
A tiling count is meaningful only for a precisely defined board. This teaching board is the ordinary Aztec diamond, not either augmented rectangle in the paper.
- Board cells
- 24
- Dominoes
- 12
- Exact tilings
- 64
The order-3 ordinary Aztec diamond has 64 domino tilings. The paper’s augmented rectangles are different regions.
Baseline check: T(Aztec diamond of order n) = 2ⁿ⁽ⁿ⁺¹⁾/²
TRY THIS
Move the tiling index while keeping the order fixed. The arrangement changes, but the total number of legal arrangements does not.
What this experiment represents. Exact enumeration of ordinary Aztec diamonds of orders 1–4. These are pedagogical baselines, not Propp’s augmented or punctured regions and not a reproduction of the proposed kernel theorem.
Mathematics · Preprint
Catalan Kernels for Augmented Aztec Rectangles
Weighted enumeration of the two families in Propp’s Problem 29
Candidate resolutionMF-PRISM-MATH-2026-14 / v1.2
Develops weighted enumeration formulas for augmented Aztec rectangles and the centrally punctured family. Catalan–Toeplitz and sparse-polynomial kernels replace large graph determinants with explicit size-reduced expressions.
- Identifier
- MF-PRISM-MATH-2026-14
- Series
- Mathematics
- Edition
- Version 1.2
- Length
- 23 pages
- Reserved DOI
- 10.5281/zenodo.22728338 (record reserved)
01
Plain-language guide
3 minute explanation
What this paper is really saying.
The paper replaces enormous domino-matching determinants with much smaller Catalan and sparse-polynomial kernels for the two augmented Aztec-rectangle families in Propp’s Problem 29.
domino tilingsaztec rectanglescatalan kernels
Counting domino tilings can be converted into a determinant, but the natural determinant grows with the full region and quickly becomes unwieldy.
The paper compresses the calculation. For the unpunctured and centrally punctured families, it derives structured kernels whose size is tied to the augmentation rather than the entire graph, making the formulas more explicit and the computations easier to audit.
02
How it works
The argument, without the notation.
The paper’s technical details matter, but the basic route can be understood in three moves.
Reduce to a kernel
Eliminate determinant-one blocks to obtain Catalan–Toeplitz or sparse-polynomial kernels of much smaller size.
Handle the puncture
Track seam parity, central deletion, boundary conditions, and normalization for the punctured family.
03
What it adds
The useful takeaways.
Explicit weighted formulas are proposed for both families named in Propp’s Problem 29.
Large graph determinants are reduced to structured, size-compressed kernels.
The punctured case is treated separately rather than assumed to follow automatically from the unpunctured formula.
04
Scope & limits
What this does—and does not—establish.
Current status
Candidate enumeration theorems for the specified unpunctured and centrally punctured families.
This manuscript presents a candidate resolution. Publication is not verification: independent specialist review remains necessary before the result should be treated as established.
The result is restricted to the precisely defined augmented Aztec-rectangle families.
Puncture seam parity and signed-matching normalization are essential, not cosmetic details.
Equivalent formulas may exist in prior literature; publication-priority review remains necessary.
05
Why it matters
Why anyone should care.
The proposed kernels turn very large matching determinants into smaller, structured expressions that are easier to analyze and verify.
06
Key terms
The vocabulary, decoded.
These are the terms needed to understand the claim. The full paper uses them more precisely.
Domino tiling
A covering of a region by 1-by-2 or 2-by-1 dominoes with no overlaps or gaps.
Perfect matching
A selection of graph edges that covers every vertex exactly once.
Kernel
A smaller matrix or operator that preserves the quantity needed from a larger system.
Toeplitz matrix
A matrix whose entries are constant along each diagonal.
07
Independent review
Where scrutiny should concentrate.
The strongest review is not a general reaction. It tests the steps most capable of changing the conclusion.
- 01Puncture seam parity, boundary conditions, and normalization of signed matchings.
- 02Complete kernel construction, sparse minors, and determinant-one eliminations.
- 03The exact Propp family definitions, OEIS index shift, and possible equivalent prior enumerations.
Publication record
Go from explanation to evidence.
This page is a reading guide, not a substitute for the manuscript. The public record links the explanation to the paper, source package, review materials, and persistent identifier.
- Document
- Preprint
- Review status
- Open for independent review
- Published
- 2026-09-12
- DOI status
- Reserved · 10.5281/zenodo.22728338
- Canonical record
- MF-PRISM-MATH-2026-14
Metriq PRISM Laboratory, Catalan Kernels for Augmented Aztec Rectangles, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-14, Version 1.2, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.22728338.