Polygon Dissections via Lucas-Inspired Encoding

Abstract

Classical enumeration of triangulations and angulations of convex polygons is governed by the Catalan and Fuss–Catalan families. In this paper, we introduce a Lucas-inspired symbolic encoding framework for a restricted subclass of triangulations, called Lucas-compatible triangulations. The purpose of the framework is not to replace classical Catalan enumeration, but to provide a complementary structural layer that records admissible local reductions through two canonical operations. Within this restricted setting, the geometric objects remain Catalan-based, whereas the associated encoding space satisfies a Fibonacci-type recurrence. We formalize the reduction model, define admissible Lucas words, and prove structural properties of the encoding map. We further present recursive generation algorithms, analyze their output-sensitive complexity, and compare the size of the encoding space with the size of the full triangulation space. In addition, we discuss geometric constraints, equivalence phenomena, and potential uses of the encoding in compact representation, constrained enumeration, and recursion-guided generation of polygon dissections. Computational experiments support the theoretical predictions and illustrate how the proposed encoding yields a compressed symbolic view of a restricted but mathematically meaningful class of dissections.

1. Introduction

Polygon dissections most notably triangulations and their generalizations to $(m+2)$-angulations form a classical and pervasive object in enumerative combinatorics, computational geometry, and algebraic structures such as cluster algebras and generalized associahedra [1]. A triangulation of a convex polygon is obtained by inserting non-crossing diagonals until all interior faces are triangles, while an $(m+2)$-angulation partitions a polygon with $mn+2$ vertices into n disjoint $(m+2)$-gons [2]. The enumeration of these structures has been understood since the 19th century and is governed by the Catalan numbers and their multi-parameter extension, the Fuss–Catalan numbers. These sequences appear in over 200 combinatorial objects, including binary trees, parenthesizations, lattice paths, and various recursive structures [3].

Alongside enumeration, another natural problem is how to encode polygon dissections in a way that preserves local geometric structure while remaining suitable for recursive generation and algorithmic manipulation. Classical Catalan-based models such as binary trees, Dyck paths, and parenthesizations provide powerful global representations, but they do not directly isolate the specific one-dimensional local pivot reductions considered in the present work. This motivates the search for alternative symbolic frameworks adapted to constrained local reduction processes.

Despite this extensive classical theory, alternative encoding frameworks for polygon dissections particularly those not tied directly to Catalan recurrence remain comparatively underexplored. One family of sequences with deep combinatorial significance is the Lucas numbers, defined by the Fibonacci-type recurrence [4].

$$L_n=L_{n-1}+L_{n-2}, L_0=2, L_1=1, n\ge 2.$$

Lucas numbers arise naturally in cyclic tiling problems, constrained binary words, tree walks, and in several algebraic and geometric contexts involving symmetric recurrences [5]. While not traditionally associated with polygon triangulations, Lucas numbers possess structural features specifically their two-branch recursive decomposition that mirror certain local reduction patterns occurring in polygon dissections [6].

Recent work on Catalan-type structures, Lucas analogues, combinatorics on words, and recursive symbolic representations suggests that classical counting sequences can often coexist with alternative structural encodings. In particular, studies on Lucas analogues of Catalan objects, constrained binary words, and symbolic recursive decompositions indicate that recurrence-driven encodings may reveal structural information not visible at the level of counting formulas alone. Our contribution is situated in this line of research, but focuses specifically on polygon dissections under geometrically admissible local pivot reductions.

The central idea of this paper is to introduce a Lucas-inspired encoding framework that does not replace or contradict Catalan enumeration but instead provides a complementary structural viewpoint for understanding how dissections can be decomposed, generated, and represented. It is important to emphasize that the proposed encoding is not defined for all triangulations. Instead, it applies to a restricted class characterized by local pivot configurations. This restriction is intrinsic to the one-dimensional recursive structure of Lucas-type encoding and distinguishes our framework from classical Catalan-based decompositions. More precisely, although the number of triangulations is classical (Catalan), when triangulation is represented through constrained Lucas words, it follows Fibonacci-type recursion in the encoding space of local diagonal choices. This yields a dual combinatorial structure: geometry remains Catalan and encoding behavior exhibits Fibonacci-like branching. Such an encoding perspective is valuable for several reasons. First, it offers a compact symbolic representation of triangulations and $(m+2)$-angulations that is compatible with dynamic generation algorithms. Second, it provides new insights into recursive decomposition patterns that are not apparent from classical Catalan formulations. Third, Fibonacci-driven encodings may facilitate advances in areas such as Gray-code enumeration, random sampling, and data compression of combinatorial objects.

In this paper, we formalize this connection by defining Lucas words as symbolic sequences corresponding to two fundamental local reduction steps: an “adjacent” contraction and a “skip-one” contraction. We prove that every Lucas-compatible triangulation admits such an encoding. The extension to $(m+2)$-angulations is discussed but not fully characterized.

Building on this structure, we design algorithms that generate dissections together with their Lucas encodings and analyze their complexity and correctness.

Computational experiments further demonstrate the consistency, scalability, and structural richness of the proposed framework.

The main contributions of this paper are as follows. First, we define a restricted geometric class, namely Lucas-compatible triangulations, together with an admissible symbolic reduction model based on two canonical steps. Second, we separate geometric enumeration from encoding enumeration and show that, within the restricted class, the encoding space satisfies a Fibonacci-type recurrence while the geometric counting background remains Catalan-based. Third, we provide recursive algorithms for generating admissible encodings and Lucas-compatible triangulations, together with correctness arguments and output-sensitive complexity discussion. Fourth, we illustrate the framework computationally and discuss how the encoding may serve as a compact symbolic representation for constrained generation and structural classification.

Related Work and Positioning of the Present Study

Classical representations of polygon triangulations are closely linked to Catalan objects such as full binary trees, Dyck paths, parenthesizations, and noncrossing partitions. These models are central in enumerative combinatorics because they encode the full class of triangulations through well-studied bijective correspondences.

On the other hand, Fibonacci- and Lucas-type structures arise naturally in constrained recursive systems, cyclic binary words, tilings, and restricted combinatorial growth models. In such settings, the essential feature is often not total geometric enumeration but the existence of a two-branch local recursion with admissibility constraints.

The gap addressed in this paper lies between these two traditions. While Catalan-based models encode the full triangulation space, they do not isolate the specific reduction dynamics generated by the two local pivot moves considered here. Conversely, Lucas-type recurrences have strong symbolic and recursive behavior, but they have not been systematically used to encode a geometrically constrained subclass of polygon triangulations in the present form.

Therefore, the present study should be understood not as an alternative counting theory for all triangulations, but as a restricted structural encoding framework. Its novelty lies in coupling admissible local geometric reductions with a Fibonacci-type symbolic recursion, thereby producing a compressed representation of a nontrivial subclass of dissections.

Overall, this study shows that Lucas numbers provide a mathematically meaningful and algorithmically powerful perspective for understanding polygon dissections. By separating enumeration from encoding, we introduce a hybrid combinatorial model which preserves classical results while simultaneously revealing new structural layers. In contrast to bijective Catalan models such as binary trees or Dyck paths, the present encoding is intentionally non-bijective and restricted. Its aim is not to represent the full triangulation space, but to capture admissible one-dimensional reduction behavior in a compact symbolic form.

2. Preliminaries

The purpose of this section is to establish the mathematical foundations necessary for the Lucas-based encoding framework developed in subsequent sections, see [7,8,9,10]. We review the classical theory of polygon triangulations and $(m+2)$-angulations, summarize key properties of Catalan and Fuss–Catalan numbers, introduce Lucas numbers and their combinatorial interpretations, and formally define the symbolic encoding objects Lucas words used throughout the paper [11,12,13]. Together, these components form a unified foundation bridging traditional geometric enumeration with the structural encoding viewpoint proposed in this work.

2.1. Polygon Triangulations and Fuss–Catalan Numbers

Let $P_n$ denote a convex polygon with $n\ge 3$ vertices. A triangulation of $P_n$ is a maximal set of non-crossing diagonals that partition the polygon into exactly $n-2$ triangles [14]. The number of triangulations is one of the most classical results in combinatorics and is given by the Catalan number

$$C_{n-2}={\displaystyle \frac{1}{n-1}}\left(\begin{array}{c}2n-4\\ n-2\end{array}\right).$$

Catalan numbers appear in numerous equivalent combinatorial objects: binary trees, Dyck paths, parenthesizations, stack-sorting permutations, and non-crossing partitions [15]. This deep interconnectedness is a key reason that triangulations continue to be extensively studied.

The recursive structure behind polygon triangulations is well understood. If one fixes vertex 1 and considers all triangles incident to it, say $(1,k,k+1)$, the polygon splits into two subpolygons. This decomposition yields the recurrence:

$$C_{n-2}={\displaystyle \sum _{k=2}^{n-1}}{C}_{k-2}{C}_{n-k-1},$$

which highlights the inherently binary (two-subproblem) nature of triangulation generation.

2.2. $(m+2)$-Angulations and Fuss–Catalan Numbers

For a fixed integer $m\ge 1$, an $(m+2)$-angulation of a convex polygon with $mn+2$ vertices is a partition of the polygon into exactly n disjoint $(m+2)$-gons using non-crossing diagonals [16,17]. These objects generalize triangulations $(m=1),$ quadrangulations $(m=2)$, pentangulations $(m=3),$ and so forth. The parameter n here represents the number of interior faces (disjoint polygons) resulting from the dissection.

The number of $\left(m+2\right)$-angulations is given by the Fuss–Catalan number

$$F{C}_n^{\left(m\right)}={\displaystyle \frac{1}{mn+1}}\left(\begin{array}{c}\left(m+1\right)n\\ n\end{array}\right).$$

Fuss–Catalan numbers arise naturally in $(m+1)$-ary rooted trees, cluster algebras and generalized associahedra, higher-order polygon dissections, multi-parameter lattice paths [18,19].

This classical enumeration theory forms the geometric backdrop for our Lucas-structured encoding perspective [20].

2.3. Lucas Numbers and Their Analytical Properties

The Lucas numbers ${\left(L_n\right)}_{n\ge 0}$ are defined by the linear recurrence

$$L_n=L_{n-1}+L_{n-2}, L_0=2, L_1=1, n\ge 2.$$

They share the same recurrence as Fibonacci numbers but differ in initial conditions, leading to distinct combinatorial behaviors [21].

Closed-Form Expression (Lucas–Binet Formula)

Let $\varphi ={\displaystyle \frac{1+\sqrt{5}}{2}}$ and $\psi ={\displaystyle \frac{1-\sqrt{5}}{2}}$. Then:

$$L_n={\varphi }^n+{\psi }^n.$$

This implies the asymptotic growth

$$L_n\sim {\varphi }^n \left(n\to \infty \right),$$

which provides a useful upper bound for algorithmic analysis [22].

Lucas numbers count cyclic binary words of length $n$ with no two consecutive 1’s, circular tilings of an $n$-cell ring with monominoes and dominoes, certain restricted walks on rooted trees and directed acyclic graphs and ranked binary strings with cyclic boundary conditions.

These interpretations demonstrate that Lucas numbers often arise when a structure admits a two-option recursive expansion a property central to our encoding framework [23].

2.4. Lucas Words for Polygon Encodings

We now introduce the symbolic encoding objects that link polygon dissections with Lucas-type recursion [24].

Definition 1 (Lucas Word).

A Lucas word of length$k$ is a string $w\in {\left\{U,V\right\}}^k$ whose symbols represent two types of permissible local reduction steps:

• $U$-step (adjacent contraction): Reduction by one vertex analogous to using a diagonal that forms a triangle adjacent to the current pivot.
• $V$-step (skip-one contraction): Reduction by two vertices corresponding to a diagonal that “skips” one vertex relative to the pivot.

The underlying geometric interpretation is formalized in Section 3.

Not every binary string over $\{U,V\}$ corresponds to a valid dissection. A Lucas word is valid if each step corresponds to an admissible diagonal, V-steps occur only when the skip-two configuration does not violate convexity or crossing rules and induced reduction process terminates exactly at a triangle or $(m+2)$-gon.

We show later that every Lucas-compatible triangulation and every $(m+2)$-angulation admits at least one valid Lucas encoding, though the encoding is typically not unique [25]. It is crucial to distinguish two notions:

• Geometric enumeration: Counts triangulations by Catalan numbers and counts $(m+2)$-angulations by Fuss–Catalan numbers [26].
• Encoding enumeration: Counts the number of possible encodings of dissections, not the dissections themselves. This encoding space satisfies a Fibonacci-type recurrence due to the two-step reduction rules [13].

Thus, polygon counts remain classical, but their symbolically encoded representations follow the algebraic structure of Lucas-inspired symbolic vocabulary. This duality lies at the heart of the contributions developed in the rest of the paper [27].

3. Lucas-Compatible Structural Formulation

The purpose of this section is twofold. First, we clarify that Lucas-compatibility is a geometric admissibility condition imposed on triangulations relative to a fixed pivot vertex. Second, we distinguish the full triangulation space from the smaller class of triangulations that admit a complete recursive reduction by the two canonical local moves used in our encoding. This distinction is essential for interpreting all subsequent counting, algorithmic, and experimental results.

In this section, we formalize the mapping between polygon dissections and their symbolic representations using Lucas-based sequences. While the total number of triangulations is traditionally governed by Catalan numbers, we propose that the recursive reduction of these structures can be uniquely captured through a symbolic alphabet derived from Fibonacci-type recurrence. We denote the set of all possible triangulations for a convex $n$-gon as $T\left(n\right)$, $T_n^{LC}\subseteq T(n)$ the set of Lucas-compatible triangulations of a convex n-gon and $E_n$ we denote the set of admissible Lucas encoding words produced by the Lucas reduction process, i.e., $E_n=\left\{w\in {\left\{U,V\right\}}^{*}\right\}$ where w is obtained from valid Lucas-compatible reduction in some triangulation in $T_n^{LC}$ and U denotes an adjacent reduction step, V denotes a skip-one reduction step.

Definition 2 (Lucas Word of a Triangulation).

Let ${T\in T}_n^{LC}$ be a Lucas-compatible triangulation of a convex polygon $P_n$. A Lucas encoding word associated with $T$ is a word

$$w\left(T\right)=w_1{w}_2\dots w_k w_i\in \{\mathrm{U},\mathrm{V}\}$$

obtained by recording the sequence of reduction steps in a valid Lucas-compatible reduction of $T$, where $U$ denotes an adjacent reduction step, $V$ denotes a skip-one reduction step. The encoding word satisfies the total reduction condition

$${\displaystyle \sum _{i=1}^k}\omega \left(w_i\right)=n-3,$$

where $\omega \left(U\right)=1$ and $\omega \left(V\right)=2.$

Theorem 1 (Basic properties of admissible Lucas encodings).

Let T be a Lucas-compatible triangulation of a convex $n$-gon. Then:

(i) 

Every admissible Lucas encoding word w associated with T has total reduction weight $n-3$, where each U-step contributes 1 and each V-step contributes 2.

(ii) 

The set of admissible encoding words is a constrained subset of ${\left\{U,V\right\}}^{*}$, satisfying the total weight condition.

(iii) 

The encoding map

$$\varphi :T_n^{LC}\to {\left\{U,V\right\}}^{*}$$

in general is not injective and is not surjective onto ${\left\{U,V\right\}}^{*}$. If its codomain is restricted to $E_n :=\varphi \left(T_n^{LC}\right)$ then $\varphi$ is surjective.

Proof. 

(i) Let a be the number of $U$-steps and b the number of V-steps in an admissible Lucas-compatible reduction. A $U$-step removes one vertex and a V-step removes two vertices. Since the reduction starts with P_n and terminates at $P_3$, the total number of removed vertices is $n-3$. Hence $a+2b=n-3$. Therefore ${\displaystyle {\sum }_{i=1}^k}\omega \left(w_i\right)=a\omega (U)+b\omega (V)=a+2b=n-3$.

(ii) Let $W_n=\{w\in {\left\{U,V\right\}}^{*} | total weight\left(w\right)=n-3\}$. By part (i), every admissible word belongs to $W_n$, so $E_n\subseteq W_n$. However, admissibility is not determined only by membership in $W_n$. At each reduction stage, the prescribed $U$- or $V$-step must be geometrically feasible for the current triangulated polygon. Thus, a word of total weight $n-3$ is admissible only if it can be realized by a valid Lucas-compatible reduction sequence. Therefore $E_n$ is a geometrically constrained subset of $W_n$.

(iii) The map $\varphi$ records only the symbols $U$ and $V$, not the specific vertices, ears, or diagonals involved in the reduction. Hence different Lucas-compatible triangulations may yield the same encoding word, so $\varphi$ is generally not injective. Moreover, $\varphi$ is defined only on $T_n^{LC}$, not on the whole set of triangulations of $P_n$. As a map

$$\varphi :T_n^{LC}\to {\left\{U,V\right\}}^{*}$$

it is not surjective, since words with total weight different from $n-3$ cannot lie in its image. Finally, if $E_n :=\varphi \left(T_n^{LC}\right)$ then the restricted map $\varphi :T_n^{LC} \to E_n$ is surjective by definition of image. □

3.1. Theoretical Framework for Lucas Encoding

The Lucas-based symbolic reduction considered in this paper is not intended to represent all possible triangulations through an arbitrary pivot decomposition. Instead, we introduce a restricted recursive framework based on two canonical local pivot configurations, which we call $U$-steps and $V$-steps. This yields a well-defined symbolic encoding class for triangulations that admit such reductions. We refer to these triangulations as Lucas-compatible triangulations.

The core of our approach lies in the systematic reduction of a polygon $P_n$ by selecting a fixed pivot vertex, labeled as Vertex 1. The structural complexity of a triangulation is decomposed by identifying a valid pivot triangle incident to this vertex under the Lucas-compatible constraints.

Unlike traditional bit-string methods that often lack geometric intuition, our Lucas-inspired model utilizes the inherent recursive symmetry of the dissection process. We introduce $E_n$ as the cardinality of the encoding space, defined as the number of distinct Lucas words generated by all Lucas-compatible triangulations of an $n$-gon.

Under this restricted reduction framework, the encoding process satisfies a precise recurrence relation.

For $n \ge 5$, every admissible encoding sequence begins with either a $U$-step or a $V$-step. If the first step is a U-step, the polygon is reduced to a convex $(n-1)$-gon, and the remaining suffix of the sequence is an admissible encoding for that polygon. If the first step is a V-step, the polygon is reduced to a convex $(n-2)$-gon, and the remaining suffix is an admissible encoding for that polygon. These two cases are disjointed and exhaustive. Therefore, the sequence $\left(E_n\right)$ satisfies the recurrence $E_n=E_{n-1}+E_{n-2}.$

The initial conditions are given by $E_3= 1$ and $E_4= 1$. In the case that $n = 3$, the polygon is already a triangle, so the only admissible encoding is the empty sequence. In the case that $n = 4$, there is exactly one admissible reduction sequence. Thus, the encoding count follows the Fibonacci recurrence, and we obtain $E_n=F_{n-2},$ where $F_1 = 1$ and $F_2 = 1$ denote the standard Fibonacci numbers.

The initial conditions are given by $E_3= 1$ and $E_4= 1$, corresponding to the number of distinct Lucas-compatible reduction sequences for the triangle and quadrilateral, respectively. This formulation reflects the fact that while the geometric count of triangulations follows Catalan numbers, the combinatorial structure of the encoding process exhibits Fibonacci-type growth.

We now formalize the enumeration of admissible encoding sequences via generating functions. Let

$$A\left(x\right)={\sum }_{n\ge 3}{E}_n{x}^{n-3}$$

be the generating function of admissible encoding sequences.

Since $E_n=E_{n-1}+E_{n-2}$ with $E_3= 1$, $E_4= 1$, we obtain

$$A\left(x\right)=1+xA\left(x\right)+x^{2}A\left(x\right)$$

and hence

$$A\left(x\right)={\displaystyle \frac{1}{1-x-x^2}}.$$

Solving this functional equation yields the Fibonacci generating function, confirming that the number of admissible encodings follows Fibonacci growth.

This recurrence holds because the class of admissible sequences is closed under $U$- and $V$-extensions, meaning that every admissible prefix yields a valid reduced subproblem that preserves geometric feasibility.

Definition 3 (Lucas-compatible pivot reduction).

Let $P_n$ be a convex $n$-gon with vertices labeled $v_1, v_2, \dots , v_n$ in counterclockwise order, and let $v_1$ be the fixed pivot. We say that a triangulation $T$ of $P_n$ is Lucas-compatible if it admits a recursive reduction in which, at each stage, the triangle selected at the pivot belongs to one of the following two canonical local configurations:

i. 

(U-step) Adjacent ear reduction: If $T$ contains the ear triangle $(v_1, v_2, v_3)$, then we remove $v_2$ and replace $P_n$ by the smaller polygon $\left(v_1, v_3, \dots , v_n\right)$ which has $n-1$ vertices. This reduction contributes the symbol $U$.

ii. 

(V-step) Skip-one ear reduction: If $T$ contains the triangle $(v_1, v_3, v_4)$ and the edge $(v_1, v_3)$ separates off exactly one boundary vertex $v_2$, then we remove $v_2$ and $v_3$ and replace $P_n$ by the smaller polygon $(v_1, v_4, v_5, \dots , v_n)$, which has $n-2$ vertices. This reduction contributes the symbol $V$.

A Lucas-compatible reduction is any recursive sequence of such steps that terminates at the base polygon $P_3$. The associated Lucas word is the word over $\{U,V\}$ obtained by recording the successive reduction types. Any triangulation that does not admit such a recursive sequence is called non-Lucas-compatible.

Remark 1.

The restriction to the two canonical pivot configurations is essential. For a general pivot triangle of the form $(v_1, v_k, v_{k+1})$ with $k>3$, the triangulation typically splits into two smaller subpolygons rather than a single reduced polygon. Therefore, such a configuration does not define a one-dimensional Lucas step of size $n-1$ or $n-2$. For this reason, the present encoding framework is not formulated for arbitrary pivot triangles, but only for Lucas-compatible local reductions.

Example 1 (Complete Encoding for a Hexagon).

To provide a fully explicit illustration of the encoding process, we consider the case of a convex hexagon $(n=6).$ It is well known that the total number of triangulations is $C_4=14$. However, as discussed in Section 3.1, the Lucas encoding framework applies only to Lucas-compatible triangulations. Therefore, not all triangulations of the hexagon admit a valid Lucas word encoding. We explicitly enumerate all Lucas-compatible triangulations and their corresponding encoding sequences.

Table 1. Lucas encoding for all Lucas-compatible triangulations of a hexagon $(n=6).$

Triangulation Type	Reduction Sequence	Lucas Word
Fan triangulation	(1, 2, 3) → (1, 2, 3) → (1, 2, 3)	UUU
Type A	(1, 2, 3) → (1, 3, 4)	UV
Type B	(1, 3, 4) → (1, 2, 3)	VU

presents the complete mapping between Lucas-compatible triangulations and their associated Lucas words.

We verify completeness by enumerating all triangulations of the hexagon and checking which admit Lucas-compatible reductions. Only three triangulations satisfy the constraints, corresponding exactly to the listed sequences. Since the total number of triangulations of a hexagon is 14, this shows that only a strict subset satisfies the Lucas-compatible constraints.

For a hexagon $(n = 6)$, the reduction process must terminate at the base triangle $P_3$. Since a $U$-step reduces the number of vertices by 1 and a V-step reduces it by 2, the total reduction must equal 3. Therefore, the only admissible Lucas words are $UUU, UV,$ and $VU$.

Figure 1 illustrates three valid reduction scenarios based on the proposed $U$-step and $V$-step framework. Scenario $A \left(UUU\right)$ represents the fan triangulation, where three consecutive U-steps reduce the hexagon to the base triangle $P_3$. Scenario $B \left(UV\right)$ shows a mixed reduction, where an initial $U$-step is followed by a $V$-step, reducing the polygon from $P_6$ to $P_3$. Scenario $C \left(VU\right)$ illustrates the alternative mixed reduction, where a $V$-step is followed by a $U$-step.

In all cases, the total vertex reduction equals $n - 3 = 3$, which is a necessary condition for admissible Lucas encodings. The figure also demonstrates that only specific sequences over $\{U, V\}$ are valid, highlighting the constrained nature of the encoding space.

More generally, as illustrated in Figure 2, the Lucas-compatible reduction process is restricted to two canonical local configurations. These configurations ensure that each reduction step produces a single smaller polygon, thereby enabling a one-dimensional recursive encoding governed by a Fibonacci-type recurrence.

Proposition 1 (Immediate necessary condition for Lucas-compatibility).

A triangulation of a convex n-gon can be Lucas-compatible only if, at every stage of the recursive reduction, the fixed pivot vertex belongs to a triangle of one of the two canonical forms permitted by Definition 3. In particular, any triangulation whose pivot decomposition necessarily creates a split into two nontrivial subpolygons at some stage is not Lucas-compatible.

Proof. 

This follows directly from the definition of Lucas-compatible reduction. The encoding allows only reductions that preserve a single reduced polygon after each step. Any pivot configuration producing two independent subproblems falls outside the admissible one-dimensional reduction scheme. □

Remark 2 (Equivalent viewpoint).

Equivalently, a triangulation is Lucas-compatible if and only if there exists an ordering of its pivot-incident reductions, always taken with respect to the fixed pivot vertex 1, such that each reduction removes either one boundary vertex or two consecutive boundary vertices while preserving convexity and noncrossing structure until the base triangle is reached.

This characterization highlights that admissible reductions must preserve a single connected polygon throughout the process. As shown in Figure 3, not every pivot configuration satisfies this requirement: certain choices induce a decomposition of the triangulation into two nontrivial subpolygons (Subproblem A and Subproblem B), resulting in a branching structure rather than a single recursive reduction. Since such a decomposition no longer produces a single smaller polygon, it fails to define a one-dimensional recursive step. These configurations are therefore excluded from the Lucas-compatible framework, as they violate the structural constraints required for a Fibonacci-type encoding.

The strictness of the Lucas-compatible class is illustrated in Figure 4. For the hexagon, the full triangulation space contains $C_4=14$ triangulations, whereas only three admit the prescribed Lucas-compatible reduction sequences.

Panel A shows the three Lucas-compatible triangulations of the convex hexagon, corresponding to the admissible reduction sequences $UUU, UV$ and $VU$. Panel B presents representative non-Lucas-compatible triangulations. These examples do not admit a complete reduction sequence consisting only of the allowed U- and V-steps with respect to the fixed pivot vertex $v_1$. Thus, the hexagon already demonstrates that Lucas-compatibility defines a strict subset of the full triangulation space.

Lucas-compatibility is not determined merely by the presence of diagonals in a triangulation, but by the existence of an admissible reduction order using only $U$- and $V$-steps from the fixed pivot vertex $v_1$. The hexagon case shows this clearly: among the 14 triangulations, only 3 satisfy the Lucas-compatible reduction condition.

Lemma 1.

Every Lucas-compatible triangulation of a convex $n$-gon admits at least one valid Lucas word encoding.

Proof. 

By definition, a Lucas-compatible triangulation admits a recursive reduction in which each step is either a $U$-step or a $V$-step. Recording these steps produces a word over $\{U,V\}.$ Since each reduction decreases the number of vertices by either 1 or 2, the process terminates after finitely many steps at the base triangle $P_3$. Hence the resulting word is a valid Lucas encoding. □

3.2. Structural Remarks on the Restricted Class

• The class is nonempty for all n ≥ 3 because fan triangulations are always Lucas-compatible.
• The class is a strict subset of all triangulations for n ≥ 6, as shown already by the hexagon example.
• Full characterization of the class remains open, but the reduction-based definition yields effective recognition and generation procedures.

Definition 4 (Encoding-equivalence).

Two Lucas-compatible triangulations are said to be encoding-equivalent if they admit at least one common admissible Lucas word. This defines an equivalence relation on the set of Lucas-compatible triangulations after fixing a choice rule for admissible words, or more generally a natural partition into fibers of the encoding map.

These fibers measure the loss of geometric information under symbolic compression and provide a natural way to study non-injectivity of the encoding.

4. Algorithmic Framework

This section presents the algorithmic foundations of our Lucas-based encoding system for polygon triangulations and $(m+2)$-angulations. We develop constructive procedures that jointly generate geometric dissection (triangulation or angulation) and the corresponding Lucas word encoding.

These algorithms demonstrate explicitly how the Fibonacci-type recurrence arises as a structural property of the encoding space, even though the set of geometric objects remains governed by Catalan or Fuss–Catalan enumeration. The section is divided into three parts: fundamental principles, triangulation algorithms, and generalization to $(m+2)$-angulations.

4.1. Fundamental Principles of Lucas-Based Algorithm Design

The Fibonacci-type recursion arises from two geometrically meaningful reduction operations:

• Adjacent reduction (“U-step”): Selecting a diagonal that reduces the polygon from n vertices to $n-1$ vertices.
• Skip-one reduction (“V-step”): Selecting a diagonal that reduces the polygon from n vertices to $n-2$ vertices.

These two reduction mechanisms represent a natural binary branching of the recursive generation tree. We have the geometric recursion $\mid T\left(n\right)\mid =C_{n-2}$. Thus, our algorithms operate in a dual combinatorial landscape: Catalan-type branching in geometry and Fibonacci-type branching in encoding.

This duality is the conceptual basis for the algorithms below.

We now present a recursive algorithm that generates all admissible reduction sequences for $P_n$.

Since the number of admissible sequences follows Fibonacci growth, the complexity can be bounded accordingly. The time complexity of Algorithm 1 is $O\left(F_n\right),$ where $F_n$ is the Fibonacci number, since each recursive branch corresponds to a Fibonacci-type expansion.

Algorithm 1. Generate Encodings(n)

Input: Integer $\mathit{n}\ge 3$ Output: Set of admissible reduction sequences $\mathcal{L}\left(\mathit{n}\right)$ Procedure:            If $\mathit{n}=3$:          return $\left\{\mathit{\epsilon }\right\}$ 2.            If $\mathit{n}=4$:          return $\left\{\mathit{U}\right\}$ 3.            Otherwise:          ${\mathit{S}}_1=\{\mathit{U}+\mathit{w}|\mathit{w}\in \mathit{G}\mathit{e}\mathit{n}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{E}\mathit{n}\mathit{c}\mathit{o}\mathit{d}\mathit{i}\mathit{n}\mathit{g}\mathit{s}(\mathit{n}-1)\}$          ${\mathit{S}}_2=\{\mathit{V}+\mathit{w}|\mathit{w}\in \mathit{G}\mathit{e}\mathit{n}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{E}\mathit{n}\mathit{c}\mathit{o}\mathit{d}\mathit{i}\mathit{n}\mathit{g}\mathit{s}(\mathit{n}-2)\}$ 4.            Return ${\mathit{S}}_1\cup {\mathit{S}}_2$

Output-Sensitive Complexity and Memory Usage

The relevant complexity measure in the present setting is output-sensitive complexity, because the algorithms are designed to enumerate admissible encodings or Lucas-compatible triangulations rather than decide mere existence. Accordingly, the running time should be compared not with the full Catalan triangulation space, but with the size of the restricted output family actually generated.

For encoding generation, the recurrence mirrors the Fibonacci growth of the admissible word set. Therefore, even before implementation details are considered, the asymptotic running time cannot be better than linear in the number of generated encodings. With memoization, repeated subproblems are avoided and the recursion tree becomes consistent with the size of the realized encoding family.

For triangulation generation, the algorithm is likewise output-sensitive: its running time is proportional to the number of Lucas-compatible triangulations produced, up to polynomial bookkeeping factors associated with storing diagonals, checking admissibility, and concatenating partial encodings.

This algorithm generates all admissible reduction sequences. To construct triangulations, we extend the reduction process to operate on geometric structures.

Proposition 2.

Let $E_n$ denote the number of admissible Lucas words for the convex n-gon. Then Algorithm 1 runs in $O\left(E_n\right)$ recursive output steps, and with memoization it uses O(n + E_n) space to store recursion states and generated words. If word construction cost is counted explicitly, the complexity is $O\left({\sum }_{w\in E_n}\left|w\right|\right)$, which is still output-sensitive.

The Algorithm 2 runs in time proportional to the number of Lucas-compatible triangulations generated.

Algorithm 2. Generate Lucas Compatible Triangulations(n)

Input: A convex polygon ${\mathit{P}}_{\mathit{n}}$ Output: Set of Lucas-compatible triangulations with encodings Procedure:            If $\mathit{n}=3$:              return the unique triangulation with encoding ε            For each admissible reduction step:           If triangle (1,2,3) exists:          apply U-step and recurse on Pn−1           If triangle (1,2,3) satisfies skip-one condition:          apply V-step and recurse on Pn−2            Concatenate symbols with recursive outputs            Return all constructed pairs (T,W(T))

Proposition 3.

Let L_n denote the number of Lucas-compatible triangulations of the convex n-gon. Then Algorithm 2 has output-sensitive running time O(L_n⋅p(n)), where p(n) is a polynomial factor accounting for admissibility tests, maintenance of polygon structure, and symbolic concatenation.

Since L_n is typically much smaller than the Catalan number C_n−2, the algorithm may provide a substantial reduction in search space when the goal is to generate only triangulations compatible with the Lucas reduction scheme.

4.2. Correctness of the Algorithms

Theorem 2.

Algorithm 1 generates exactly all admissible reduction sequences.

Proof. 

Each valid sequence must begin with either $U$ or $V$. These correspond to reductions to $P_{n-1}$ and $P_{n-2}$, respectively. The recursion exhaustively generates all valid sequences without overlap. □

Theorem 3.

Algorithm 2 generates all Lucas-compatible triangulations together with valid reduction encodings.

Proof. 

The algorithm explores all admissible pivot configurations defined in Section 3.1. Each step preserves non-crossing constraints and reduces the polygon size. Since all admissible reductions are considered recursively, the algorithm generates exactly all Lucas-compatible triangulations. □

The proposed framework encodes only Lucas-compatible triangulations, which form a strict subset of all triangulations. Thus, not all triangulations admit a valid encoding, the encoding is not unique and the mapping is not bijective. The encoding should therefore be interpreted as a structural compression of admissible reduction patterns rather than a complete enumeration of triangulations.

The present framework does not provide a bijective encoding of all triangulations, nor does it compete with classical Catalan bijections at the level of full enumeration. Its role is different: it identifies a geometrically meaningful restricted class for which one-dimensional admissible reductions induce a compact symbolic structure. In this sense, the framework is best viewed as a constrained structural encoding rather than a universal representation theorem.

The reduction framework can be generalized to $(m+2)$-angulations by defining analogous local reduction steps. However, the exact enumeration of admissible encoding sequences in the generalized setting depends on additional geometric constraints and is not necessarily governed by a Lucas-type or Fibonacci-type closed form. Therefore, we restrict our formal counting results to triangulations and leave the full characterization of the generalized case for future work.

4.3. Why the Extension to $(m+2)$-Angulations Is More Difficult

The main obstacle in extending the framework from triangulations to $(m+2)$-angulations is that the reduction process is no longer governed by only two canonical local moves. In the triangulation case, the U- and V-steps produce a one-dimensional recursive reduction. In the general angulation case, admissible pivot configurations may depend on face size, may remove larger boundary blocks, and may generate branching patterns that are not captured by a second-order Fibonacci-type recurrence. For this reason, the present paper restricts its formal recurrence and complexity results to triangulations, while treating angulations as a promising but technically richer direction for future work.

5. Computational Experiments

This section presents an experimental validation of the reduction-based encoding framework introduced in Section 3 and Section 4. Our primary objective is to verify the correctness of admissible reduction sequences and to analyze their growth relative to classical Catalan enumeration. Beyond validating the recurrence, the experiments are designed to quantify the gap between geometric complexity and encoding complexity, illustrate non-injectivity, and evaluate the potential of the encoding as a compact representation of a restricted triangulation family.

All experiments were conducted using Python 3.12 on a standard workstation (Intel i7 CPU, 16 GB RAM, Ubuntu 22.04). The implementation follows the recursive definitions of Algorithms 1 and 2 without heuristic pruning.

Memorization was used to avoid redundant computations.

For each $n \in \{3, 4, \dots , 12\}$, we generated the complete set of admissible reduction sequences $\mathcal{L}\left(n\right)$ using Algorithm 1. We verified that the number of admissible sequences satisfies $E_n=E_{n-1}+E_{n-2}$ with initial conditions $E_3= 1$ and $E_4= 1$. The computed values are given in

Table 2. Comparison between full triangulation counts and admissible encoding counts for convex n-gons.

$\mathit{n}$	${\mathit{E}}_{\mathit{n}}$	${\mathit{C}}_{\mathit{n}-2}$	Sequences	$\mathbf{Compression} \mathbf{Ratio} {\mathit{E}}_{\mathit{n}}/{\mathit{C}}_{\mathit{n}-2}$
3	1	1	$\epsilon$	1.000
4	1	2	$U$	0.500
5	2	5	$UU,V$	0.400
6	3	14	$UUU,UV,VU$	0.214
7	5	42	…	0.119
8	8	132	…	0.061
9	13	429	…	0.030
10	21	1430	…	0.015

.

These results confirm that the encoding space follows a Fibonacci-type growth $E_n= F_{n-2}.$ We compared the number of triangulations $\mid T\left(n\right)\mid =C_{n-2}$ with the number of admissible encodings $E_n$.

The results in Figure 5 show that Catalan growth is exponential, encoding growth is Fibonacci-type and significantly slower. The scripts used to generate the figures are provided in the Supplementary Materials. This demonstrates that the encoding space forms a strict subset of all triangulations and provides a compressed structural representation.

Table 3. Properties of the Mapping from Triangulations to Reduction Sequences.

Property	Observation	Interpretation
Multiple triangulations → same encoding	Distinct triangulations may realize the same admissible Lucas word	The encoding map is gene-rally non-injective
Partial domain	Not every triangulation admits a complete Lucas-compatible reduction	The encoding is defined only on a restricted subclass
Realized encoding family	Only geometrically admissible words are retained	The symbolic space is constrained by geometric feasibility
Structural abstraction	The encoding records reduction patterns rather than full geometry	The framework is a compressed structural representation

summarizes the structural properties of the mapping between triangulations and reduction sequences.

Figure 6 compares the geometric complexity and encoding complexity for polygon triangulations. The Catalan counts represent the number of triangulations, whereas the encoding counts represent the number of admissible reduction sequences. The bar chart shows that the encoding space remains significantly smaller than the geometric space for all tested values of $n$, confirming the reduced computational overhead of the reduction-based encoding framework.

The logarithmic bar chart highlights the structural gap between geometric and encoding complexity. While the number of triangulations grows according to the Catalan sequence, the number of admissible reduction sequences follows a Fibonacci-type recurrence. This demonstrates that the encoding framework does not enumerate all triangulations, but rather a restricted family of admissible reduction paths, leading to a substantially smaller search space and improved computational efficiency.

The experimental results validate the theoretical model: the encoding space follows a Fibonacci-type recurrence, the geometric space remains Catalan-based, and the framework provides meaningful structural compression. These findings support the interpretation of Lucas-inspired reduction encoding as a constrained symbolic representation of polygon dissections.

Let

$$r_n={\displaystyle \frac{\left|T_n^{LC}\right|}{C_{n-2}}}$$

denote the proportion of Lucas-compatible triangulations. Experimental data for the tested range suggest that the proportion r_n decreases rapidly with n, which is consistent with the conjecture that Lucas-compatible triangulations form an asymptotically sparse subset of all triangulations.

5.1. Compression and Structural Gap

The main experimental observation is not merely that the encoding count follows a Fibonacci-type recurrence, but that the admissible encoding space grows substantially more slowly than the full triangulation space. This confirms that the proposed framework acts as a structural filter: it retains only triangulations compatible with the local Lucas reduction scheme and represents them through a much smaller symbolic family. From a computational point of view, this reduced state space is relevant whenever the target task concerns constrained generation rather than complete Catalan enumeration.

5.2. Non-Injectivity and Equivalence Behavior

To illustrate the non-injective nature of the encoding map, we examined small instances in which distinct Lucas-compatible triangulations admit the same admissible word. These examples show that the encoding captures reduction behavior rather than full geometric identity. In this sense, the encoding should be interpreted as a structural abstraction: triangulations that differ geometrically may still belong to the same encoding fiber if they realize the same pivot-reduction pattern.

Figure 7 demonstrates that the Lucas encoding does not uniquely determine a triangulation: different triangulations can yield identical admissible encoding sequences, revealing an inherent non-injectivity in the encoding map.

Two distinct triangulations $T_A$ and $T_B$ of the hexagon are shown, both admitting the same admissible encoding sequence $VU$ with respect to the fixed pivot vertex $v_1$. In each case, the first step removes an internal triangle not incident to $v_1$ (a $V$-step), reducing the polygon from six to four vertices. The second step removes an ear triangle incident to $v_1$ (a $U$-step), yielding the base triangle. Although the reduction sequences coincide, the initial triangulations differ, demonstrating that the encoding map is not injective.

5.3. Practical Implications of the Encoding

Although the present study is theoretical, the experiments suggest three practical implications. First, the encoding offers a compact symbolic representation of admissible reduction patterns. Second, it may support constrained enumeration and recursion-guided sampling by reducing the search space from Catalan growth to Fibonacci-type encoding growth within the restricted class. Third, it provides a natural indexing layer for grouping triangulations by reduction behavior, which may be useful in compression, classification, and algorithmic preprocessing.

These observations admit an algorithmic interpretation. As illustrated in Figure 8, Lucas-compatible encodings can be used as a control mechanism for recursion-guided generation of constrained triangulations, thereby bridging the gap between the combinatorial framework and potential applications. Starting from a polygon with a fixed pivot vertex $v_1$, candidate encoding sequences are first enumerated subject to the feasibility condition $a+2b=n-3$. Each sequence then guides a recursive construction process, where at each step a valid removal ($U$ or $V$) is selected in accordance with the constraints. Constraint checking and early pruning eliminate infeasible branches, ensuring that only valid partial structures are extended. The process terminates when the base triangle is reached, yielding a set of valid triangulations. The workflow highlights potential applications in areas such as computer graphics and computational biology.

6. Conclusions and Future Work

This study introduced a unified structural and algorithmic framework that connects polygon triangulations and $(m+2)$-angulations with Lucas-inspired symbolic encodings. While the number of dissections remains governed by the classical Catalan and Fuss–Catalan sequences, we demonstrated that the encoding space of valid reduction patterns follows a Fibonacci-type recurrence. This reveals a clear structural duality: geometric objects are Catalan-based, whereas their symbolic representations exhibit Fibonacci-type recursive behavior.

We derive recurrence-based expressions for the number of admissible encodings and analyze their structural properties. On the algorithmic side, we designed constructive procedures that simultaneously generate polygon dissections and their Lucas encodings. We proved correctness, analyzed computational complexity, and confirmed optimal performance relative to the inherent combinatorial growth of these objects. Computational experiments validated all theoretical predictions, illustrating the consistency between geometric enumeration and symbolic Lucas-based decomposition. The results further suggest that Lucas encodings may provide a compact symbolic representation within the restricted class of polygon dissections considered here, particularly for larger polygons. The framework does not aim to replace Catalan enumeration but rather provides a constrained symbolic layer capturing a subset of recursive reduction patterns.

Overall, the framework developed here provides a coherent symbolic perspective on classical polygon dissections and establishes a foundation for future exploration of recurrence-driven encoding systems. An interesting direction for future work is to characterize the exact combinatorial structure of the Lucas-compatible class and determine its asymptotic behavior.