Di Francesco Conjecture

• Di Francesco Conjecture is a product formula that exactly enumerates domino tilings of Aztec triangles, revealing deep links with alternating sign matrices.
• The proof evolved from computer-intensive determinant evaluations to a concise combinatorial argument using factorization and complementation theorems.
• Its significance lies in exposing hidden symmetry in tiling enumeration, bridging ideas from Aztec diamonds to more complex cruciform regions.

Searching arXiv for papers on Di Francesco's conjecture, especially the Aztec triangle formulation and its proofs. The Di Francesco conjecture most commonly denotes a 2021 conjectural product formula for the number of domino tilings of the Aztec triangle ${\mathcal T}_n$, a family of hybrid square-lattice regions combining part of a square with half of an Aztec diamond. In its now-proved form, the conjecture states that the tiling number is

$$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$

where $M(R)$ denotes the number of domino tilings of a region $R$. The formula is explicitly described as reminiscent of the product formula for alternating sign matrices, and its proof history is notable for the transition from computer-intensive methods to a short combinatorial argument based on perfect-matchings factorization and complementation theorems (Byun et al., 6 Aug 2025).

1. Statement and geometric definition

For each positive integer $n$, the region ${\mathcal T}_n$ is an Aztec triangle of order $n$. It is obtained by starting with a square of side-length $2n$, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order $n-1$. In the more explicit construction recorded in the literature, one starts with a $2n\times 2n$ square, forms the upper portion above the zigzag cut, and then places the upper half of $$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$0 above it, right-justified (Ciucu, 2021).

The conjectured, and now established, enumeration formula is

$$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$1

Di Francesco’s formulation was immediately striking because of its proximity to the alternating sign matrix product formula

$$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$2

The resemblance is not merely aesthetic: the Aztec triangle sits combinatorially between classical Aztec-diamond tiling enumeration and objects linked to alternating sign matrices, and the product structure suggested a hidden symmetric mechanism rather than an ad hoc determinant identity (Ciucu, 2021).

2. Structural context: Aztec triangles, cruciform regions, and divisibility

A major step toward the conjecture was the introduction of cruciform regions $$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$3, which generalize Aztec diamonds by growing them in four directions. These regions are built by superimposing two Aztec rectangles, producing a cross-shaped domain with four piers whose lengths are governed by parameters $$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$4. The tileability condition recorded for these regions is

$$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$5

For balanced cruciform regions, the number of domino tilings admits a closed hyperfactorial product formula (Ciucu, 2021): $$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$6 where $$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$7, and

$$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$8

This cruciform formula yielded partial progress toward Di Francesco’s conjecture. In particular, $$M({\mathcal T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!},$$9 arises as a fundamental sector inside a symmetrized cruciform region, and the divisibility relation

$M(R)$0

was established. The same work also derived product formulas for related elbow regions and showed that the Aztec-triangle count consistently divides the tiling count of larger symmetric regions with explicit formulas (Ciucu, 2021).

This divisibility phenomenon strongly suggested that the Aztec-triangle formula should be accessible through symmetry-breaking operations on more tractable symmetric graphs. A plausible implication is that the conjecture was structurally “close” to known factorization mechanisms, even before a direct proof existed.

3. Why the conjecture was difficult

Although the product formula is compact, proving it turned out to be unusually resistant to direct argument. The 2025 proof emphasizes that the conjecture was “a real challenge to prove without the use of computers,” and that the two earlier proofs—one due to Koutschan, Krattenthaler and Schlosser, and one due to Corteel, Huang and Krattenthaler—both relied on substantial computer calculations that would be hard to check directly (Byun et al., 6 Aug 2025).

One computational route passed through determinant evaluation. In work inspired by Di Francesco’s determinant for twenty-vertex configurations, a determinant identity was proved whose $M(R)$1 specialization is equivalent to the Aztec-triangle product formula. The determinant

$M(R)$2

was shown to evaluate to

$M(R)$3

and the underlying determinants were described as counting twenty-vertex model configurations, or equivalently the number of domino tilings of Aztec triangles (Koutschan et al., 2024).

That route confirms the conjectural formula, but it does so through holonomic and determinant-evaluation machinery rather than through a direct combinatorial decomposition of the tilings. The distinction matters because the later short proof isolates the underlying symmetry mechanisms.

4. Short combinatorial proof

The 2025 proof gives a short and elementary combinatorial proof based on two standard but powerful tools in perfect-matching enumeration: the factorization theorem for perfect matchings of symmetric planar bipartite graphs and the complementation theorem for perfect matchings of cellular graphs (Byun et al., 6 Aug 2025).

The first step is the standard translation from domino tilings to perfect matchings of planar dual graphs. The argument then compares the Aztec triangles with larger cruciform and nearly-cruciform regions whose matching numbers are accessible by product formulas. A key identity obtained in the reduction is

$M(R)$4

where the denominator is a nearly-cruciform region.

The proof “undoes” a symmetrization in three steps. Repeated use of the factorization theorem reduces symmetric planar graphs to appropriate half-graphs, each reduction contributing a power of $M(R)$5. The complementation theorem is then applied recursively to the nearly-cruciform regions until one reaches a boundary region described as a trimmed Aztec rectangle with intrusions. At that stage, the tilings can be determined by classical formulas, including the Lindström–Gessel–Viennot lemma or Aztec rectangle theorems (Byun et al., 6 Aug 2025).

The recursive simplification yields the ratio

$M(R)$6

This ratio matches the ratio predicted by the product formula, and together with the initial value at $M(R)$7 gives the conjecture for all $M(R)$8 by induction (Byun et al., 6 Aug 2025).

5. Conceptual significance

The conjecture is significant less for the bare existence of a product formula than for the type of structure it exposes. Aztec triangles are not classical symmetric regions like Aztec diamonds; they are hybrid objects obtained by splicing a zigzag-cut square with half of an Aztec diamond. Despite that asymmetry, their tiling numbers still factor in a way “reminiscent” of alternating sign matrix enumeration (Ciucu, 2021).

The proof clarifies that the product formula is governed by symmetry at the level of the dual matching graph, rather than by obvious symmetry of the original region. The crucial tools all act naturally on perfect matchings:

Ingredient	Role
Domino tilings $M(R)$9 perfect matchings	Reformulates the problem graph-theoretically
Factorization theorem	Splits symmetric planar bipartite graphs into halves
Complementation theorem	Reduces cellular graphs while controlling matching counts
Cruciform product formulas	Supplies explicit multiplicative evaluations

This suggests that Aztec triangles belong to a broader class of regions whose enumeration becomes tractable after suitable symmetrization and then controlled symmetry-breaking. A plausible implication is that the product form is a manifestation of a hidden factorization pattern rather than an isolated coincidence.

6. Terminological ambiguity and other “Di Francesco conjectures”

The expression “Di Francesco conjecture” is not unique across the literature. In algebraic geometry, the Di Francesco–Itzykson conjecture concerns the polynomiality of Severi degrees for plane curves with a fixed number of nodes. One paper computed node polynomials up to $R$0, proved universal formulas for the first nine coefficients, and verified the optimal-threshold conjecture up to $R$1 (Block, 2010). Related work proved the Di Francesco–Itzykson–Göttsche shape conjectures for node polynomials of $R$2, expressing the counts through Bell polynomials in universal linear forms in the Chern numbers (Qviller, 2010).

In another combinatorial direction, the name refers to a conjecture on permutation matrices and descending plane partitions without special parts. A bijection was constructed that preserves the quadruple of statistics considered by Behrend, Di Francesco and Zinn–Justin, thereby proving that conjecture in the case without special parts (Fulmek, 2018).

There is also a Di Francesco conjecture connected with generalized lambda-determinants and $R$3 T-systems. Solutions with principal coefficients were expressed combinatorially via partition functions of perfect matchings, non-intersecting paths, and networks, and the resulting framework specializes to the lambda-determinant setting (Vichitkunakorn, 2015).

For that reason, in current combinatorics the unqualified phrase Di Francesco conjecture is best interpreted contextually. In the tilings literature, it most often denotes the Aztec-triangle product formula proved combinatorially in 2025 (Byun et al., 6 Aug 2025).