---
title: Triangle & Quadrilateral Counting Maps
url: https://www.emergentmind.com/topics/triangle-and-quadrilateral-counting-maps
type: topic
---

# Triangle & Quadrilateral Counting Maps

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 [1901.04981].

## 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 [1901.04981].

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.

Source: https://www.emergentmind.com/topics/triangle-and-quadrilateral-counting-maps