A short combinatorial proof of Di Francesco's conjecture on Aztec triangles

Mihai Ciucu; Seok Hyun Byun

REVIEW 3 major objections 4 minor 1 cited by

A short combinatorial proof of Di Francesco's conjecture on Aztec triangles

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves the conjectured product formula for the number of domino tilings of the n-th Aztec triangle—$2^{n(n-1)/2}$ times the product of $(4i+2)!/(n+2i+1)!$—by a short combinatorial argument that cancels matching counts of identical

desk verdict A genuinely new proof of a known formula; the new nearly-cruciform product formula is the real payoff, but the cancellations lean heavily on visual graph identifications. read the letter →

arxiv 2508.04545 v1 pith:OT4KHVTJ submitted 2025-08-06 math.CO cond-mat.stat-mechmath-phmath.MP

Seok Hyun Byun , Mihai Ciucu This is my paper · ORCID

classification math.COcond-mat.stat-mechmath-phmath.MP MSC 05A1505B4505C70

keywords Aztectriangledominotilingsperfectmatchingsfactorizationtheoremcomplementationcruciformgraphsproductformulacombinatorialproof

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 proves the 2021 conjecture on Aztec triangles: the number of domino tilings of the n-th Aztec triangle equals $2^{n(n-1)/2}$ times the product of $(4i+2)!/(n+2i+1)!$ for $i=0$ to $n-1$. The proof is a short chain of applications of two known results on perfect matchings—the factorization theorem for symmetric graphs and the complementation theorem—together with a new product formula for matchings of a family of nearly-cruciform graphs. The authors introduce a 'cancellation trick': the same factor graph appears in the factorization of two different symmetric graphs, so dividing the equations cancels those factors and relates the Aztec triangle tiling counts to ratios of easily computed quantities. The argument avoids the large determinant evaluations and computer algebra on which the two earlier proofs relied.

What carries the argument

The factorization theorem for perfect matchings of symmetric planar bipartite graphs, which writes the matching count of a symmetric graph as $2^k$ times the product of the matching counts of its two halves. The proof applies it three times and uses the fact that the same graph $F_n$ (respectively $B_n$, $S_n$) appears in two different factorizations, so dividing the identities cancels that factor. The second new component is the nearly-cruciform graph $D^{a,b,a,d}_{m,n}$ and its product formula (2.3), proved via the complementation theorem and the trimmed Aztec rectangle enumeration (Lemma 3.1).

What would settle it

For a fixed small n, such as n=2, explicitly describe the vertex and edge sets of the graph $F_n$ obtained from equation (2.4) and of the graph $F_n$ obtained from equation (2.5), and check whether they are literally the same graph; a single case where they differ would invalidate the division leading to equation (2.6).

Watch

Extended reading notes

Core claim

The central discovery is that the sequence $M(T_n)$ is determined by the ratio recurrence $M(T_{n+1})/M(T_n) = M(C^{n+1,n,n,n}_{2n+1,2n+1}) / (2 M(D^{n,n,n,n}_{2n+1,2n+1}))$, where $C$ is a cruciform graph (whose matchings are given by Theorem 2.1) and $D$ is a new nearly-cruciform graph with a product formula (Theorem 2.2). Combining these formulas gives $M(T_{n+1})/M(T_n) = 2^n n!!/(3n)!! \cdot (4n+2)!/(3n+2)!$, which matches the ratio of the claimed expressions for $M(T_{n+1})$ and $M(T_n)$, and the base case $n=1$ checks. The new element is the product formula (2.3) for nearly-cruciform graphs with symmetric pier lengths, derived from the complementation theorem and the trimmed Aztec rec

Load-bearing premise

The cancellation steps treat the graph $F_n$ produced in equation (2.4) as literally the same graph as the $F_n$ produced in (2.5) (similarly for $B_n$ and $S_n$), based on visual comparison rather than an explicit vertex-by-vertex bijection; if any of these identifications misses a difference, the divisions that yield equations (2.6), (2.10), and (2.13) would be invalid.

Editorial extensions

If this is right

Formula (1.1) for the Aztec triangle tiling numbers now stands on a proof that can be checked by hand, without computer algebra.
The ratio recurrence $M(T_{n+1})/M(T_n) = M(C^{n+1,n,n,n}_{2n+1,2n+1}) / (2 M(D^{n,n,n,n}_{2n+1,2n+1}))$ reduces the conjecture to known formulas for cruciform and nearly-cruciform graphs.
Theorem 2.2 gives a new closed formula for the number of perfect matchings of balanced nearly-cruciform graphs with equal opposite pier lengths.
Combined with the base case $M(T_1)=1$, the ratio recurrence determines $M(T_n)$ for every $n$, confirming the prediction from the twenty-vertex model.

Reading between the lines

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

The cancellation trick used here—dividing two factorization identities that share the same factor graph—may apply to other symmetric graph families whose matching counts are known, producing ratio recurrences without determinant evaluations.
The chain of equalities suggests that, up to powers of two, Aztec triangle tilings could be paired with tilings of smaller regions; making this pairing explicit could yield a fully bijective proof of (1.1).
Theorem 2.2's product formula holds only for symmetric pier lengths; the arithmetic complexity for asymmetric nearly-cruciform graphs points to a hypergeometric evaluation that might also have a closed form.
The simplified recurrence may clarify the connection to the twenty-vertex model by isolating the factor $(4i+2)!/(n+2i+1)!$ as the ratio of two matching counts.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main enumeration (1.1) was already established by two computer-assisted proofs, so the headline result is not new. What is new is the proof strategy: a clean three-step symmetrize-factor-cancel argument using Ciucu's factorization and complementation theorems, plus a new product formula (Theorem 2.2) for nearly-cruciform graphs with a=c. That theorem is the real payload. If it holds, it is a solid addition to the program from [3].

The paper is well organized and honest. It does not use the target formula as an input; it uses the ratio of (1.1) only to verify the final product, and the base case n=1. No circularity. The authors are explicit about what is new and what is borrowed, and the proof of Theorem 2.2 follows the same route as [3] but ends in a graph covered by the Elkies–Gessel lemma. That is a legitimate contribution.

The soft spots are exactly where the reader's stress-test lands. In (2.4)-(2.5), (2.8)-(2.9), and (2.11)-(2.12), the cancellation depends on the claim that two graphs arising from separate applications of the factorization theorem are literally the same graph, not merely isomorphic. The text says "precisely the same" and points to figures. For a reader checking line by line, that is a gap. A referee should ask for explicit vertex/edge identifications or coordinate descriptions. Similarly, the exponent t = n-2a-2 in (3.3) is stated after "a straightforward analysis" and is not fully expanded. These are not fatal: the pictures are consistent and the cited theorems are published, but this is exactly where a subtle sign error would hide.

The citation pattern is fine. Self-citation is appropriate since the machinery is the second author's own. The abstract honestly notes that the prior proofs used heavy computer calculations and that this one does not.

Who is this for? Combinatorialists working on perfect matchings, factorization theorems, and Aztec-type regions. It deserves a serious referee, not a desk reject. If I were editor, I would send it to a referee familiar with [1],[2],[3] and ask that the graph identifications and the exponent calculation be written out. Even with those gaps, the core argument is credible and the new theorem is interesting enough to justify peer review.

Referee Report

3 major / 4 minor

Summary. The paper proves Di Francesco's 2021 conjecture that the number of domino tilings of the Aztec triangle T_n equals 2^{n(n-1)/2} times the product over i=0 to n-1 of (4i+2)!/(n+2i+1)!. The proof is entirely combinatorial: it introduces a new family of nearly-cruciform graphs and proves a product formula for them (Theorem 2.2) using Ciucu's complementation theorem and factorization theorem, then builds a chain of three cancellation steps that relate ratios of matching counts of various cruciform graphs to the ratio M(T_{n+1})/M(T_n). The final ratio is evaluated using Theorems 2.1 and 2.2 and matched against the ratio predicted by the conjecture; the base case n=1 verifies the formula. The argument avoids the computer calculations of the two previous proofs.

Significance. If the proof is fully fleshed out, this is a significant contribution: it gives a short conceptual proof of a conjecture that previously required substantial computer-assisted verification, and it introduces a new exactly solvable family of graphs with a product formula. The paper does not use the conjectured formula as an input; the final comparison is a genuine verification of the ratio. The strengths are the clear architectural idea, the reduction to published theorems and Lemma 3.1, and the absence of computer calculations. The main weakness is that several load-bearing graph identifications and portions of the proof of Theorem 2.2 are asserted by reference to figures rather than demonstrated.

major comments (3)

[§2, equations (2.4)–(2.13)] The cancellation chain is the heart of the proof. Equations (2.4) and (2.5) are divided after canceling M(F_n), but the only evidence that the F_n produced in the two factorizations are the same graph is the sentence 'compare the right picture in Figure 4 and the bottom right picture in Figure 3.' The same visual identification is used for B_n between (2.8) and (2.9) and for S_n between (2.11) and (2.12). Perfect matching counts are sensitive to every vertex and edge; a difference in even one edge or vertex invalidates the division. Please provide explicit vertex/edge bijections, or coordinate descriptions, for F_n, B_n, and S_n in their two occurrences, or state and prove a lemma identifying the graphs. This is load-bearing because the entire chain leading to (2.7) collapses if any of these equalities is not established.
[§3, proof of Theorem 2.2] The proof of Theorem 2.2 is only an outline. The exponent t=n-2a-2 in equation (3.3) is asserted as a 'straightforward analysis' with no calculation; the identification of D^{a+n,b-n,a+n,d-n}_{m+n,0} with the doubly-intruded Aztec rectangle ˙AR_{2n+2a+1,m+n}(n-d,n-b) is justified visually; and the factorization step (3.7) states without derivation that both G+ and G- are AR-type graphs. Since Theorem 2.2 is a new result and supplies the exact value used in equation (2.14), these omissions are load-bearing. Please expand the complementation calculation, define the map from the final nearly-cruciform graph to the Aztec rectangle graph, and derive (3.7) explicitly.
[§3, Theorem 2.2 statement vs. proof] Theorem 2.2 is stated for D^{a,b,a,d}_{m,n}, but the proof begins by applying the complementation theorem to 'D^{a,b,a,b}_{m,n}' and equation (3.3) concludes with D^{a+1,b-1,a+1,b-1}_{m+1,n-1}; the next equation (3.4) uses d-i. This internal inconsistency makes it unclear whether the theorem is proved for arbitrary d or only for d=b. Since the application in this paper needs only d=b, one fix is to state Theorem 2.2 in that special case; if the general d statement is intended, correct the proof. As written, the proof of the stated theorem is not self-consistent.

minor comments (4)

[§2, notation] The graph with the dot (e.g., ˙C) is defined once but the dot is sometimes typeset ambiguously. Also, in equation (2.14) the denominator uses M(˙C^{n,n,n,n}_{2n+1,2n+1}) where equation (2.7) has M(D^{n,n,n,n}_{2n+1,2n+1}); this relies on M(D)=M(˙C), which should be stated at that point.
[§3, Figure 9 caption] The caption gives explicit AR_{11,13}(...) examples. It would help to also display the removed sets S and T in the notation of Lemma 3.1 for this example, so the reader can verify the pattern without reconstructing it.
[§3, Theorem 2.2 statement] The balance condition a+b+c+d=m+n-2 is stated in the preceding paragraph, but Theorem 2.2 would be easier to use if the condition (here a+b+a+d=m+n-2) were included in the theorem statement explicitly.
[§3, text around (3.3)] The occurrence 'Da,b,a,b' in the proof is either a typo for 'Da,b,a,d' or a sign that the proof only covers d=b. This should be corrected to match the theorem statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof derives the ratio from independent factorization/complementation theorems and checks the base case.

full rationale

The derivation of Di Francesco's formula (1.1) is an induction: the base case n=1 is checked, and the ratio M(T_{n+1})/M(T_n) is computed from Theorem 2.1 (an independent product formula for cruciform graphs proved in Ciucu's earlier paper [3]), the newly proved Theorem 2.2 (proved using the complementation theorem and the independent Lemma 3.1), and the factorization theorem. The conjecture (1.1) is never used as an input; the final step compares an independently computed ratio to the ratio of the two sides of (1.1) and then uses the base case. The intermediate cancellations in equations (2.6), (2.10), and (2.13) are algebraic divisions that rely on structural coincidences of the graphs F_n, B_n, and S_n. These coincidences are asserted by comparing figures rather than by explicit bijections, which is a rigor gap, but it is not circular: the graph coincidences are not derived from the conjectured formula, and no fitted parameter is renamed as a prediction. The heavy use of the second author's earlier theorems is legitimate external support: [1], [2], and [3] are published, general results that do not assume Di Francesco's conjecture. Therefore the central claim is not forced by definition, by self-citation, or by fitting.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests entirely on published theorems (factorization, complementation, Ciucu's cruciform formula, and the EKLP/Helfgott-Gessel lemma) plus a new theorem (Theorem 2.2) proved in the paper. No empirical free parameters are fitted: all parameters are structural sizes of the graphs. The new graph family (nearly-cruciform graphs) is an explicit construction and does not constitute a postulated unobservable entity.

assumptions (5)

standard math Ciucu's factorization theorem for symmetric planar bipartite graphs
Invoked to express matchings of the symmetric augmented graphs as a product of two half-graph matchings; used throughout Section 2 and in Section 3. Published in [1].
standard math Ciucu's complementation theorem for perfect matchings of graphs with a cellular completion
Used in Section 3 to successively transform the nearly-cruciform graph into an Aztec rectangle with intrusions; the special case and exponent rule are taken from [3, Section 3].
standard math Ciucu's product formula for cruciform graphs ([3], Theorem 2.1)
Provides the numerator M(C^{n+1,n,n,n}_{2n+1,2n+1}) used in (2.7) and (2.14); this is a published theorem and is not re-derived in the present paper.
standard math Elkies-Kuperberg-Larsen-Propp and Helfgott-Gessel product formula for trimmed Aztec rectangles with boundary deletions (Lemma 3.1)
Used at the end of the proof of Theorem 2.2 to evaluate M(AR_{r,s}(S)) and M(AR_{r,s}(T)).
standard math Planar dual equivalence between domino tilings of a region and perfect matchings of its dual graph
The paper's tilings are counted as perfect matchings of dual graphs; this equivalence is standard in the field.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A short combinatorial proof of Di Francesco's conjecture on Aztec triangles." pith.science (2026). https://pith.science/paper/OT4KHVTJ

@misc{pith2026250804545,
  author       = {Pith},
  title        = {Pith review of: A short combinatorial proof of Di Francesco's conjecture on Aztec triangles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OT4KHVTJ}},
  note         = {Machine review of arXiv:2508.04545}
}

read the original abstract

Di Francesco conjectured in 2021 that the number of domino tilings of a certain family of regions -- called Aztec triangles -- on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. This turned out to be a real challenge to prove without the use of computers -- each of the two existing proofs (one due to Koutschan, Krattenthaler and Schlosser, the other to Corteel, Huang and Krattenthaler) relies on substantial computer calculations which would be hard to check directly. In this paper we present a short combinatorial proof that relies on the second author's factorization theorem and complementation theorem for perfect matchings.

Figures

Figures reproduced from arXiv: 2508.04545 by the authors.

**Figure 1.** Half of the square of side length 2n, and half of the Aztec diamond of order n − 1, for n = 5 (left). The region Tn for n = 5 (center), and its planar dual graph (right). symbolic parameter. As the authors state, due to their size, the results of the intermediate calculations cannot be displayed in their article (and are shown instead in an accompanying electronic material [13]). The other existing proof of formula … view at source ↗

**Figure 2.** Left. The cruciform graph C a,b,c,d m,n for m = 9, n = 6, a = 3, b = 4, c = 5 and d = 2. Right. The graph C˙ n,n,n,n 2n+1,2n+1 has the same number of perfect matchings as the near-cruciform graph D n,n,n,n 2n+1,2n+1. Here and throughout the paper, circled edges indicate forced edges (i.e., edges that are contained in every perfect matching). The cruciform graph C a,b,c,d m,n is the subgraph of the square grid descri… view at source ↗

**Figure 3.** [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 3.** Figure 3: Obtaining equation (2.4) (here n = 3). Top left. ... C n+1,n+1,n+1,n+1 2n+1,2n+1 is obtained from the cruciform graph C n+1,n+1,n+1,n+1 2n+1,2n+1 by removing the three indicated boundary vertices. Top right. Due to forced edges, ... C n+1,n+1,n+1,n+1 2n+1,2n+1 has the…

**Figure 4.** Figure 4: Left. The graph C˙ n,n,n,n 2n+1,2n+1 for n = 3. Right. Applying the factorization theorem to a graph with twice as many perfect matchings as C˙ n,n,n,n 2n+1,2n+1 produces the graph En on top and the same graph Fn on the bottom as on the bottom right in [PITH_FULL_IMA…

**Figure 5.** Figure 5: Left. Obtaining equation (2.8) (here n = 3). When applying the factorization theorem to a graph having twice as many matchings as En, the resulting G+ is the graph An, while the resulting G− is the graph Bn. Right. Obtaining equation (2.9) (here n = 3). When applying t…

**Figure 6.** Figure 6: Left. Obtaining equation (2.11) (here n = 3). When applying the factorization theorem to the graph An, the resulting G+ is the Aztec triangle graph Tn+1, while the resulting G− is the graph Sn. Right. Obtaining equation (2.12) (here n = 3). When applying the factorizat…

**Figure 7.** Figure 7: Successive applications of the complementation theorem to the nearly-cruciform graphs: D 3,3,3,3 7,7 yields D 3+1,3−1,3+1,3−1 7+1,7−1 , i.e. D 4,2,4,2 8,6 (top left); D 4,2,4,2 8,6 yields D 4+1,2−1,4+1,2−1 8+1,6−1 , i.e. D 5,1,5,1 9,5 (top right); D 5,1,5,1 9,5 yields …

**Figure 8.** Figure 8: Three more applications of the complementation theorem (top left, top right and bottom left); the circles in the bottom left figure mark missing lattice points from the final graph. The evolution of the boundary of the initial nearly-cruciform graph (bounded by the bl…

**Figure 9.** Figure 9: Left. D a+n,b−n,a+n,d−n m+n,0 has the same number of matchings as AR˙ 2n+2a+1,m+n(n − d, n − b), the shown doubly-intruded Aztec rectangle with top left vertex removed. Right. When we apply the factorization theorem to a graph with twice as many matchings as AR˙ 2n+2a…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Disentangling electronic and phononic contributions to high-temperature superconductivity in X2MH6 hydrides
cond-mat.supr-con 2026-04 unverdicted novelty 4.0 of 10

In the X2MH6 hydride family, electronic contributions (X–H distance, H ELF networking, H-projected DOS at EF) dominate Tc under isoelectronic substitution, while pressure trades electronic gain against phononic loss.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

[3]

Ciucu, Cruciform regions and a conjecture of Di Francesco, Proc

M. Ciucu, Cruciform regions and a conjecture of Di Francesco, Proc. Amer. Math. Soc. 150 (2022), 4655– 4670

work page 2022
[1]

Ciucu, Enumeration of perfect matchings in graphs with reﬂec tive symmetry, J

M. Ciucu, Enumeration of perfect matchings in graphs with reﬂec tive symmetry, J. Combin. Theory Ser. A 77 (1997), 67–97

work page 1997
[2]

Ciucu, A complementation theorem for perfect matchings of g raphs having a cellular completion, J

M. Ciucu, A complementation theorem for perfect matchings of g raphs having a cellular completion, J. Combin. Theory Ser. A 81 (1997), 34––68

work page 1997
[4]

Domino tilings of generalized Aztec triangles

S. Corteel, F. Huang, C. Krattenthaler, Domino tilings of genera lized Aztec triangles, preprint, arXiv:2305.01774. 8We use here the notation [ n] = {1, . . . , n}. A SHORT COMBINATORIAL PROOF OF DI FRANCESCO’S CONJECTURE ON AZTEC TRIANGLES 13

work page Pith review arXiv
[5]

Elkies, G

M. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sig n matrices and domino tilings (Part 1), J. Alg. Combin. 1 (1992), 111–132

work page 1992
[6]

Di Francesco and E

P. Di Francesco and E. Guitter, Twenty-vertex model with dom ain wall boundaries and domino tilings, Electron. J. Combin. 27 (2020), no. 2, Paper No. 2.13

work page 2020
[7]

Di Francesco, Twenty vertex model and domino tilings of the A ztec triangle, Electron

P. Di Francesco, Twenty vertex model and domino tilings of the A ztec triangle, Electron. J. Combin. 28 (2021), Paper No. 4.38, 50

work page 2021
[8]

Fischer, Short proof of the ASM theorem avoiding the six-ver tex model, J

I. Fischer, Short proof of the ASM theorem avoiding the six-ver tex model, J. Combin. Theory Ser. A 144 (2016), 139–156

work page 2016

Show all 17 references

[9]

Helfgott and I

H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds an d hexagons with defects, Electron. J. Combin. 6 (1999), Research Paper 16, 26

1999
[10]

Koutschan, Advanced applications of the holonomic systems approach, Ph.D

C. Koutschan, Advanced applications of the holonomic systems approach, Ph.D. thesis, RISC, J. Kepler University, Linz, 2009

2009
[11]

Koutschan, Announcement, 9th International Conferen ce on Lattice Path Combinatorics and Applica- tions, CIRM Luminy, June 2021

C. Koutschan, Announcement, 9th International Conferen ce on Lattice Path Combinatorics and Applica- tions, CIRM Luminy, June 2021

2021
[12]

Koutschan, C

C. Koutschan, C. Krattenthaler, M. Schlosser, Determinant evaluations inspired by Di Francesco’s deter- minant for twenty-vertex conﬁgurations, J. Symb. Comput. 127 (2025), 102352

2025
[13]

D eterminant evaluations in- spired by Di Francesco’s determinant for twenty-vertex conﬁgu rations

C. Koutschan, Electronic material accompanying the article “D eterminant evaluations in- spired by Di Francesco’s determinant for twenty-vertex conﬁgu rations”, 2025. Available at http://www.koutschan.de/data/det3/

2025
[14]

Krattenthaler, Schur function identities and the number of perfect matchings of holey Aztec rectangles, Contemp

C. Krattenthaler, Schur function identities and the number of perfect matchings of holey Aztec rectangles, Contemp. Math. 254 (2000), 335–350

2000
[15]

Kuperberg, Another proof of the alternating-sign matrix c onjecture, Int

G. Kuperberg, Another proof of the alternating-sign matrix c onjecture, Int. Math. Res. Not. 3 (1996), 139–150

1996
[16]

Zeilberger, Proof of the alternating sign matrix conjecture Electron

D. Zeilberger, Proof of the alternating sign matrix conjecture Electron. J. Combin. 3 (1996), 1–84

1996
[17]

Zeilberger, The holonomic ansatz II

D. Zeilberger, The holonomic ansatz II. Automatic discovery(!) and proof(!!) of holonomic determinant evaluations, Annals of Combinatorics 11 (2007), 241–247. Email address : sbyun@clemson.edu School of Mathematical and Statistical Sciences, Clemson U niversity, Clemson, South...

2007

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [3]

Ciucu, Cruciform regions and a conjecture of Di Francesco, Proc

M. Ciucu, Cruciform regions and a conjecture of Di Francesco, Proc. Amer. Math. Soc. 150 (2022), 4655– 4670

work page 2022

[2] [1]

Ciucu, Enumeration of perfect matchings in graphs with reﬂec tive symmetry, J

M. Ciucu, Enumeration of perfect matchings in graphs with reﬂec tive symmetry, J. Combin. Theory Ser. A 77 (1997), 67–97

work page 1997

[3] [2]

Ciucu, A complementation theorem for perfect matchings of g raphs having a cellular completion, J

M. Ciucu, A complementation theorem for perfect matchings of g raphs having a cellular completion, J. Combin. Theory Ser. A 81 (1997), 34––68

work page 1997

[4] [4]

Domino tilings of generalized Aztec triangles

S. Corteel, F. Huang, C. Krattenthaler, Domino tilings of genera lized Aztec triangles, preprint, arXiv:2305.01774. 8We use here the notation [ n] = {1, . . . , n}. A SHORT COMBINATORIAL PROOF OF DI FRANCESCO’S CONJECTURE ON AZTEC TRIANGLES 13

work page Pith review arXiv

[5] [5]

Elkies, G

M. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sig n matrices and domino tilings (Part 1), J. Alg. Combin. 1 (1992), 111–132

work page 1992

[6] [6]

Di Francesco and E

P. Di Francesco and E. Guitter, Twenty-vertex model with dom ain wall boundaries and domino tilings, Electron. J. Combin. 27 (2020), no. 2, Paper No. 2.13

work page 2020

[7] [7]

Di Francesco, Twenty vertex model and domino tilings of the A ztec triangle, Electron

P. Di Francesco, Twenty vertex model and domino tilings of the A ztec triangle, Electron. J. Combin. 28 (2021), Paper No. 4.38, 50

work page 2021

[8] [8]

Fischer, Short proof of the ASM theorem avoiding the six-ver tex model, J

I. Fischer, Short proof of the ASM theorem avoiding the six-ver tex model, J. Combin. Theory Ser. A 144 (2016), 139–156

work page 2016