Triangle & Quadrilateral Counting Maps

• Triangle and quadrilateral counting maps are precise frameworks for enumerating planar maps decorated with embedded trees.
• The Fredes–Sepúlveda bijection transforms tree-decorated maps into simple-boundary maps, enabling explicit formulas for triangulations and quadrangulations.
• These results provide refined combinatorial and probabilistic insights, linking asymptotic behaviors to the universal Brownian map.

A triangle and quadrilateral counting map refers to a precise enumeration framework for planar maps whose faces are all triangles (triangulations, or 3-angulations) or quadrilaterals (quadrangulations, or 4-angulations), in which each map is further decorated by a specific embedded subtree. These counts are closely tied to bijective correspondences between such tree-decorated maps and simple-boundary maps, leading to explicit formulae for the number of possible configurations with prescribed numbers of faces and tree-size parameters. This framework, notably formalized via the Fredes–Sepúlveda bijection, enables refined combinatorial and probabilistic analysis of planar maps and their scaling limits (Fredes et al., 2019).

1. Definitions and Notational Foundations

A planar map is a connected graph embedded in the sphere, considered up to homeomorphism, with one distinguished oriented edge known as the root. The set $M_f$ consists of all rooted maps with exactly $f$ faces. A $q$-angulation is a planar map in which every face has degree $q$; specifically, a triangulation is a 3-angulation and a quadrangulation is a 4-angulation.

A planted plane tree of size $m$ corresponds to a rooted map with a single face and $m$ edges; denote this set as $T_m$. The cardinality $\lvert T_m\rvert = C_m$ is the $m$th Catalan number: $C_m = \frac{1}{m+1} \binom{2m}{m}.$

A tree-decorated map is a pair $(M, T)$, where $M \in M_f$ and $T \subset M$ is a (spanning or non-spanning) submap isomorphic to a planted plane tree of size $k$. The set of such maps is

$$\mathcal{M}_{f, k} = \{ (M, T) : M \in M_f,\ T \subset M,\ |T|=k\}.$$

A map with a simple boundary is a connected planar map with a distinguished external face whose boundary is a cyclic, non-self-intersecting curve of even length $2m$. The set $SB_{f, m}$ denotes rooted planar maps with $f$ interior faces and a simple boundary of length $2m$.

2. The Fredes–Sepúlveda Bijection

The central structural result is the Fredes–Sepúlveda bijection, which asserts for $f, m \ge 0$ a natural correspondence

$$\phi: \mathcal{M}_{f,m} \longleftrightarrow T_m \times SB_{f,m}.$$

This bijection operates by "ungluing" a tree-decorated map $(M, T)$ along the edges of the decorating tree $T$. Each edge of $T$ is split to become part of a new simple boundary of length $2m$ in a map $M^{\mathrm{bnd}}$, while the remainder of $M$ is unchanged. The inverse "gluing" operation reconstructs $(M, T)$ by identifying the boundary edges of $M^{\mathrm{bnd}}$ according to the contour traversal of $T$.

A key property is that, under the uniform measure on $\mathcal{M}_{f, m}$, the decorating tree $T$ is itself uniformly sampled from $T_m$.

3. Enumeration Formulae for Tree-decorated Planar Maps

Let $T_{n,k}$ denote the number of tree-decorated triangulations with $n$ faces and tree of size $k$, and $Q_{n,k}$ the number of analogous quadrangulations. By leveraging the bijection and established enumeration of simple-boundary $q$-angulations (Mullin–Krikun for triangulations, Brown–Ganaras–Miermont for quadrangulations), one obtains:

Triangulations $(q=3)$

For $0 \leq k \leq n/2+1$: $$T_{n,k} = 2^{n-2k} \frac{ \left( \frac{3n}{2} + k - 2 \right)!! } { \left( \frac{n}{2}-k+1 \right)! \left( \frac{n}{2}+3k \right)!! } \times \frac{3n}{k+1} \binom{4k}{2k,\ k,\ k}$$ where for odd integer $m$, $m!! = m\, (m-2)\, (m-4)\, \ldots\, 1$.

Quadrangulations $(q=4)$

For $0 \leq k \leq n+1$: $$Q_{n,k} = 3^{n-k} \frac{ (2n + k - 1)! }{ (n+2k)! (n-k+1)! } \times \frac{4n}{k+1} \binom{3k}{k,\ k,\ k}$$

These enumerations can also be organized into bivariate generating functions: $$T(x, y) = \sum_{n, k \geq 0} T_{n, k} x^n y^k, \quad Q(x, y) = \sum_{n, k \geq 0} Q_{n, k} x^n y^k,$$ though the above closed-form expressions are the operational descriptions.

4. Specialization to Classical Enumeration $(k=0)$

Setting $k=0$ recovers the classical enumeration of rooted $q$-angulations:

Map Type	Enumeration Formula
Triangulations	$$T_{n,0} = 2^n \dfrac{ \left( \frac{3n}{2}-2 \right)!! } { \left( \frac n2+1 \right)! \left( \frac n2 \right)!! } \times 3n$$
Quadrangulations	$$Q_{n,0} = 3^n \dfrac{ (2n-1)! }{ n!(n+1)! } \times 4n = \dfrac{ 4n }{ (n+1)(n+2) } \binom{3n}{n, n, n}$$

The triangulation formula coincides with Tutte’s enumeration of rooted triangulations of size $n$. The quadrangulation formula agrees with the standard result for rooted quadrangulations.

5. Asymptotic Behavior and Probabilistic Consequences

By applying singularity analysis to the counting expressions, one retrieves the universal cubic-root asymptotic regime for large $n$, with fixed tree-size $k$: $$T_{n,k} \sim \mathsf{c}_k\, n^{-5/2} \left( \frac{256}{27} \right)^n, \qquad Q_{n,k} \sim \mathsf{d}_k\, n^{-5/2} 12^n,$$ where $\mathsf{c}_k, \mathsf{d}_k$ are explicit positive constants.

For each fixed $k$, tree-decorated triangulations and quadrangulations reside in the same Gromov–Hausdorff–Prokhorov universality class as the pure Brownian map.

A direct corollary is that, for a uniformly chosen tree-decorated $q$-angulation with $n$ faces and tree of size $k$, the decorating tree is itself uniform in $T_k$, generalizing the uniform-tree property of spanning-tree decorated maps.

6. Broader Implications and Generalizations

The bijective framework for tree-decorated $q$-angulations extends to more general decorated planar maps and provides a unified combinatorial perspective for analyzing substructure-connectivity in random environments. The explicit enumeration not only refines the classical catalog of planar map enumeration but also enables fine-grained probabilistic investigations, particularly regarding local limits and scaling phenomena in random planar geometry (Fredes et al., 2019).

This establishes a precise connection between the combinatorics of embedded trees and the geometry of planar maps, facilitating further studies in random geometry and related probabilistic models.