Aztec Triangles in Domino Tiling Models

• Aztec triangles are square-lattice regions whose domino tilings are enumerated by a closed product formula derived through combinatorial proofs.
• They arise from a hybrid construction combining the top half of an Aztec diamond with half-square regions, and have realizations via perfect matchings and nonintersecting lattice paths.
• Their enumeration parallels alternating sign matrices and connects to the 20 Vertex model, revealing deep links between dimer models and integrable systems.

Searching arXiv for recent and foundational papers on Aztec triangles and closely related Aztec-region tilings. arXiv_search(query="Aztec triangles domino tilings Di Francesco", max_results=10, sort_by="relevance") Searching more broadly for generalized Aztec triangles and related enumeration papers. arXiv_search(query="generalized Aztec triangles domino tilings", max_results=10, sort_by="relevance") Searching for the 20-vertex-model connection to Aztec triangles. arXiv_search(query="Twenty Vertex model domino tilings Aztec triangle", max_results=10, sort_by="relevance") Aztec triangles are square-lattice regions studied through domino tilings, perfect matchings, and lattice-path correspondences. In the sense that has become central in recent work, the Aztec triangle $\mathcal{T}_n$ is a hybrid between the Aztec diamond and half-square regions, and its number of domino tilings is

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

This product formula, conjectured by Di Francesco and later established by several proofs, is reminiscent of the enumeration formula for alternating sign matrices and places Aztec triangles at an intersection of enumerative combinatorics, dimer models, and integrable lattice models (Byun et al., 6 Aug 2025).

1. Terminology and geometric scope

The supplied literature uses the term “Aztec triangle” for more than one family of regions. In current enumerative work, the dominant meaning is Di Francesco’s region $\mathcal{T}_n$, but older and adjacent literatures also use the term for half-diamond or triangle-like Aztec subregions.

Usage	Description	Source
Di Francesco’s Aztec triangle $\mathcal{T}_n$	Constructed by joining the “top half” of an Aztec diamond of order $n-1$ to the “top half” of a $2n\times 2n$ square, separated by a specific two-step zigzag cut	(Byun et al., 6 Aug 2025)
Generalized Aztec triangles	Parameterized by a partition $\mu=(\mu_1,\mu_2,\ldots,\mu_n)$ and an integer $\ell\in\{2n,2n+1\}$; the classical case is $\mu=(n,n-1,\ldots,1)$, $\ell=2n$	(Corteel et al., 2023)
Aztec half-diamond / Aztec Triangle	A subregion $$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$0 defined by explicit inequalities inside the Aztec diamond; visually “half” the Aztec diamond, bounded along a diagonal	(Nordenstam et al., 2011)
Triangle-like generalized Douglas specialization	In generalized Douglas regions, triangle-shaped specializations arise in a way described elsewhere as Aztec triangles	(Lai, 2013)

This multiplicity of usage matters because exact formulas, bijections, and asymptotic claims are not automatically interchangeable across these conventions. The modern literature on Di Francesco’s $$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$1 is primarily concerned with a specific domino-tiling enumeration problem, whereas the half-diamond literature emphasizes shuffling dynamics and limit shapes.

2. Exact enumeration and the Di Francesco conjecture

Di Francesco conjectured in 2021 that the number of domino tilings of $$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$2 is given by the product formula

$$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$3

and this same quantity was conjectured to count configurations of the 20 Vertex model on certain domains with domain wall type boundary conditions. The first values recorded for the common enumeration are

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

(Francesco, 2021).

The formula is explicitly parallel to the alternating sign matrix product

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

and that resemblance has been one of the main reasons Aztec triangles attracted attention (Byun et al., 6 Aug 2025). In the generalized framework of Aztec triangles, closed-form product formulas were proved for wider families indexed by partitions, and the classical staircase choice $$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$6 recovers Di Francesco’s conjectured formula as a special case (Corteel et al., 2023).

The proof history is itself a notable part of the subject. Earlier proofs of the classical formula relied on substantial computer calculations, specifically via the holonomic ansatz or factor exhaustion. By contrast, Byun and Ciucu later gave a short combinatorial proof described as the first short, combinatorial, computer-free proof, based on the factorization theorem and complementation theorem for perfect matchings (Byun et al., 6 Aug 2025).

3. Combinatorial realizations

A central feature of Aztec triangles is the coexistence of several exact combinatorial models for the same counting problem. In the perfect-matching framework, domino tilings are counted by passing to the dual planar graph and enumerating perfect matchings. In the generalized framework, tilings are encoded by sequences of partitions

$$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$7

subject to strip conditions: odd steps add horizontal strips, even steps add vertical strips, $$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$8 is empty, and $$M(\mathcal{T}_n)=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}.$$9 (Corteel et al., 2023).

The same objects admit tableau and path descriptions. The generalized theory gives bijections with super symplectic semistandard tableaux and with tuples of nonintersecting lattice paths. For the path model, the endpoints are

$\mathcal{T}_n$0

and the allowed steps are north, east, and northeast; in the $\mathcal{T}_n$1 case one obtains H-Delannoy paths, characterized by avoidance of a specified forbidden point (Corteel et al., 2023).

For Di Francesco’s classical triangles, a closely related description uses non-intersecting Schröder paths. Each path has right, down, and diagonal steps, and the domino-tiling problem becomes an LGV determinant problem on these path families. This path viewpoint is also the bridge to the 20 Vertex model, where boundary conditions force comparable ensembles of osculating Schröder paths (Francesco, 2021).

These parallel realizations explain why the subject supports determinant formulas, product evaluations, and structural comparisons with other triangular arrays and vertex models. A plausible implication is that the geometric shape of the region is only one layer of the theory; the more rigid content lies in the interlacing and nonintersection constraints shared by the associated path and partition systems.

4. Determinants, factorization, and proof architectures

The generalized theory of Aztec triangles is organized around determinant evaluations. For even $\mathcal{T}_n$2, the number of domino tilings is

$\mathcal{T}_n$3

where $\mathcal{T}_n$4 is the Delannoy number. For odd $\mathcal{T}_n$5, the count is

$\mathcal{T}_n$6

where $\mathcal{T}_n$7 counts H-Delannoy paths (Corteel et al., 2023). For staircase-type partitions $\mathcal{T}_n$8, these determinants can be evaluated in closed product form, producing explicit formulas $\mathcal{T}_n$9 and $\mathcal{T}_n$0. The classical Aztec triangle corresponds to the staircase partition of full length.

The short combinatorial proof of the classical formula uses a different architecture. Its starting point is a recursion for the ratio

$\mathcal{T}_n$1

where $\mathcal{T}_n$2 and $\mathcal{T}_n$3 are balanced cruciform and nearly-cruciform graphs. The argument then applies Ciucu’s factorization theorem

$\mathcal{T}_n$4

for symmetric planar bipartite graphs and the complementation theorem

$\mathcal{T}_n$5

for appropriate cellular subgraphs (Byun et al., 6 Aug 2025).

Repeated factorization and complementation reduce the problem to matchings of doubly-intruded trimmed Aztec rectangles with deleted boundary vertices, whose matching numbers are known. One explicit formula used at the end of this reduction is

$\mathcal{T}_n$6

where $\mathcal{T}_n$7. The ratio then telescopes over $\mathcal{T}_n$8, and the initial value is verified directly, yielding the product formula for $\mathcal{T}_n$9 (Byun et al., 6 Aug 2025).

5. Relation to integrable vertex models

Aztec triangles are closely linked to the 20 Vertex model. A principal result is that the number of 20 Vertex configurations on certain domains with domain wall type boundary conditions equals the number of domino tilings of the corresponding Aztec triangle. In that setting,

$n-1$0

with $n-1$1 the 20V count and $n-1$2 the domino-tiling count (Francesco, 2021).

The proof uses integrability. The 20V model is related to a six-vertex model with U-turn boundaries and specific spectral parameter values $n-1$3, which converts the partition function to a determinant. On the 20V side,

$n-1$4

with

$n-1$5

On the domino-tiling side,

$n-1$6

with

$n-1$7

The equality of the two determinants is established through an infinite lower-triangular matrix $n-1$8 with diagonal entries $n-1$9, satisfying $2n\times 2n$0 for the corresponding infinite matrices (Francesco, 2021).

This relation explains why Aztec triangles appear simultaneously in dimer enumeration and in integrable statistical mechanics. It also clarifies an important limitation: while the counts coincide, no explicit bijection is currently known between 20V configurations and domino tilings of the generalized Aztec triangles (Corteel et al., 2023).

6. Related regions, asymptotics, and broader context

Aztec triangles belong to a larger ecosystem of Aztec regions. Generalized Aztec triangles extend the classical family by partition data $2n\times 2n$1 and parity parameter $2n\times 2n$2, while trimmed Aztec rectangles, quartered Aztec rectangles, expanded Aztec diamonds, and generalized Douglas regions supply comparison objects and reduction targets in exact enumeration (Corteel et al., 2023). In particular, the combinatorial proof of Di Francesco’s conjecture proceeds through cruciform and nearly-cruciform graphs and eventually through trimmed Aztec rectangles, illustrating that Aztec triangles are embedded in a broader exact-solvability program for planar matchings (Byun et al., 6 Aug 2025).

Asymptotic analysis is more developed for related Aztec domains than for Di Francesco’s triangles themselves. For the Aztec diamond, random domino tilings exhibit an arctic phenomenon separating frozen and disordered regions, with limit shape

$2n\times 2n$3

and boundary fluctuations governed at one-point level by the Tracy–Widom distribution and process level by the Airy$2n\times 2n$4 process; the top $2n\times 2n$5 paths are conjectured to converge to the Airy line ensemble (Debin et al., 2023). The same source states that arctic phenomena and similar limit shape fluctuations are expected for a range of domino tiling domains, including Aztec triangles and more general polygons.

A different asymptotic picture appears in the half-diamond convention for “Aztec triangle.” There, the half-diamond or Aztec Triangle inherits its limit shape by restriction from the Aztec diamond, and the resulting frozen boundary is an arctic parabola rather than a circle (Nordenstam et al., 2011). This terminological split is mathematically consequential: statements about an arctic parabola pertain to the half-diamond model, whereas the modern exact product formula

$2n\times 2n$6

pertains to Di Francesco’s $2n\times 2n$7.

The current state of the subject therefore combines a highly explicit exact-enumerative theory with a less complete asymptotic theory. This suggests that one of the natural next directions is to develop for Di Francesco’s Aztec triangles an analogue of the detailed fluctuation and arctic-curve theory already available for Aztec diamonds.