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Enumeration of Corona for Lozenge Tilings

T0 review · 0 major / 3 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read Exact closed formulas for corona counts in lozenge tilings are proven via weighted adjacency matrices.

desk verdict The paper proves two OEIS conjectures on lozenge corona tilings via explicit weighted adjacency matrices and gives a parameterized generalization. read the letter →

arxiv 2504.19269 v1 submitted 2025-04-27 math.CO

Craig Knecht , Feihu Liu , Guoce Xin This is my paper
classification math.CO
keywords lozengetilingscoronaenumerationweightedadjacencymatrixperfectmatchingsclosed-formformulasOEISconjecturesdualgraph
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes closed-form expressions for the number of lozenge tilings of two specific corona-shaped regions. These expressions had been conjectured from computational evidence but lacked formal proof. The authors interpret the tilings as perfect matchings and extract the exact count from the weighted adjacency matrix of the dual graph. The same matrix method is then shown to apply in a more general setting beyond the original two cases.

What carries the argument

The weighted adjacency matrix of the dual graph, whose determinant encodes the generating function for valid lozenge placements.

What would settle it

Brute-force enumeration of all lozenge tilings of one specific corona region and direct comparison with the numerical value given by the closed formula derived from the matrix.

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Extended reading notes

Core claim

The number of lozenge tilings of the corona regions equals a specific closed formula obtained from the weighted adjacency matrix of the dual graph. This confirms the conjectures listed as A380346 and A380416 and extends the enumeration technique to broader families of corona configurations.

Load-bearing premise

The weighted adjacency matrix of the dual graph for each corona region encodes precisely the valid lozenge placements and produces the correct generating function without omitted configurations or extraneous factors.

Editorial extensions

If this is right

  • The conjectures for sequences A380346 and A380416 become proven theorems.
  • The matrix method supplies exact counts for larger corona instances without listing every tiling.
  • The approach carries over to a wider class of symmetric corona regions in lozenge tilings.
  • Algebraic extraction from the adjacency matrix replaces case-by-case combinatorial casework.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same matrix construction may resolve additional open conjectures on tiling numbers for other symmetric regions.
  • The generalization hints at a uniform algebraic pattern that could link enumerations across different families of tiling regions.
  • Physical dimer models on planar graphs could supply interpretations or asymptotic formulas for these exact counts.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper proves the conjectured closed-form enumerations for lozenge tilings of two specific corona regions (OEIS A380346 and A380416) by constructing explicit weighted adjacency matrices of the associated dual graphs and extracting the generating functions from their determinants (or appropriate evaluations). It further extends the same weighting and matrix-construction technique to a parameterized family of generalized corona regions.

Significance. If the matrix encodings are faithful, the work supplies the first rigorous proofs of these OEIS conjectures and demonstrates a reusable algebraic-combinatorial method for corona-type tiling problems. Such exact formulas are of interest in dimer models and plane-partition enumeration; the explicit, verifiable matrix constructions constitute a concrete strength of the manuscript.

minor comments (3)
  1. §2: The precise definition of the weighting scheme on the dual graph (especially the assignment of variables to boundary edges) should be stated as a numbered definition or displayed equation for easier reference.
  2. §4, Figure 2: The labeling of vertices in the generalized corona diagram is slightly inconsistent with the matrix indices used in the subsequent determinant calculation; a small table mapping labels to indices would improve readability.
  3. References: The citation to the OEIS entries should include the exact sequence numbers and the date of access or submission, following standard combinatorial-enumeration practice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of our contributions, and recommendation to accept. We are pleased that the work is recognized as providing rigorous proofs for the OEIS conjectures via explicit matrix constructions and as offering a reusable technique for generalized corona regions.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation from explicit weighted adjacency matrix is self-contained

full rationale

The paper constructs the weighted adjacency matrix directly from the dual graph of each corona region, encodes the valid lozenge placements via the chosen weights and boundary conditions, and obtains the generating function by determinant evaluation or equivalent algebraic computation. The OEIS conjectures appear solely as verification targets after the matrix is built and evaluated; they are not used to define the matrix entries or the weighting scheme. The generalization follows identically from the same construction without invoking self-citations as load-bearing premises or renaming fitted results as predictions. No step reduces the claimed count to its own input by definition or by prior author work.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The proof relies on standard linear-algebra identities for adjacency matrices of planar graphs and the correspondence between lozenge tilings and perfect matchings; no new parameters, entities, or ad-hoc assumptions are introduced beyond the usual graph-theoretic setup.

assumptions (1)
  • standard math The determinant (or suitable minor) of the weighted adjacency matrix of a bipartite graph counts the weighted perfect matchings.
    Invoked to obtain the generating function for valid corona tilings.

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Cite this review

Pith. "Pith review of Enumeration of Corona for Lozenge Tilings." pith.science (2026). https://pith.science/paper/2504.19269

@misc{pith2026250419269,
  author       = {Pith},
  title        = {Pith review of: Enumeration of Corona for Lozenge Tilings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2504.19269}},
  note         = {Machine review of arXiv:2504.19269}
}
read the original abstract

Knecht considers the enumeration of coronas. This is a counting problem for two specific types of lozenge tilings. Their exact closed formulas are conjectured in [A380346] and [A380416] on the OEIS. We prove this conjecture by using the weighted adjacency matrix. Furthermore, we extend this result to a more general setting.

Figures

Figures reproduced from arXiv: 2504.19269 by the authors.

Figure 1
Figure 1. Left-tilted, right-tilted, and vertical. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Corona of a hexagon H. The first result of this paper is to give an exact closed formula for H(n). In other words, we prove the following conjecture. Theorem 1.1. (Conjectured in [13, A380346]) Let n ∈ N. Let H(n) be the number of coronas of a hexagon H with side length n. Then there are only four cases in which the number of lozenges is used in a corona of a hexagon H with side length n, namely 6n + 3, 6n + 4, 6n +… view at source ↗
Figure 3
Figure 3. Corona of a diamond D. Our second result is to provide an exact closed formula for D(n). Theorem 1.2. (Conjectured in [13, A380416]) Let n ∈ N. Let D(n) be the number of coronas of a diamond D with side length n. Then there are only four cases in which the number of lozenges is used in a corona of a diamond D with side length n, namely 4n+ 3, 4n+ 4, 4n+ 5, and 4n + 6. Let di(n) be the number of corona tilings for 4n… view at source ↗
Figures from the paper (11 more)
Figure 13
Figure 13. Figure 13: Our third result provides an exact closed formula for the number of coronas of a [PITH_FULL_IMAGE:figures/full_fig_p003_13.png]
Figure 4
Figure 4. Figure 4: A hexagon H [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: The five states at corner 1 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: The five states at corner 2. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: The n + 1 states on the side. Now we present our proof of Theorem 1.1. Proof of Theorem 1.1. Based on the above statement, we know that corner 1 and corner 2 are connected by the n + 1 states in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: The bipartite graph in proof of Theorem 1.1. [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: A diamond D. Now we consider the corona of a diamond D. Given a diamond D with side length n, the four corners of the diamond D are denoted by 1, 2, 3, 4, see [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: The eight states at corner 1 [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: The five states at corner 2. Now we present our proof of Theorem 1.2. Proof of Theorem 1.2. Similar to the proof of Theorem 1.1, we can construct the graph in [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 12
Figure 12. Figure 12: The graph in proof of Theorem 1.2. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 13
Figure 13. Figure 13: A hexagon H and a diamond D. Given a hexagon H, a corona of a hexagon H is a lozenge tiling along the edges of H such that no additional lozenges are utilized. The number of coronas of a hexagon H is denoted by H(n1, n2, n3). Theorem 4.1. Let n1, n2, n3 ∈ N. Let H(n1,…

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