Fine Mixed Subdivisions of a Dilated Triangle

TL;DR

The paper studies fine mixed subdivisions (rhombus tilings with holey triangles) of an upward equilateral triangle of side $n$, focusing on fixed hole arrangements and how tilings change when holes are moved. It develops a spread-out based criterion for tiling existence and uniqueness, a graph-theoretic representation of the subdivision, and two families of flips—GD flips and trapezoid flips—that connect all tilings within the same hole configuration and across hole configurations, respectively. It also provides a method to identify segments forced by holes, and proves connectivity results with explicit flip-count bounds, situating the work within tropical geometry and connections to 3D tableaux. The results advance understanding of the combinatorics of rhombus tilings on triangular grids and offer tools for analyzing regularity, tiling uniqueness, and hole-movement dynamics in these systems.

Abstract

An upward equilateral triangle of side $n$ can be partitioned into $n$ unit upward equilateral triangles and $\frac{n(n-1)}{2}$ unit rhombi with $60^{\circ}$ and $120^{\circ}$ angles. In this paper, we focus on understanding such partitions with a fixed arrangement of unit triangles. We formulate a criterion for such a partition being unique, identify a set of operations that connects all such partitions, and determine which parts are common to all such partitions. Additionally, we discuss an operation that connects all possible partitions with different arrangements of triangles.

Figures (14)

• Figure 1: A fine mixed subdivision of an equilateral triangle of side length 5.
• Figure 2: A tiling with some holes whose side lengths are greater than 1.
• Figure 3: Two touching saturated triangles $s_1$ and $s_2$ and the unique way to tile $(s_1 \vee s_2)\setminus (s_1\cup s_2)$.
• Figure 4: A fine mixed subdivision with its tropical pseudoline arrangement drawn in magenta, along with its realization (in purple and orange). The purple and orange triangles correspond to $s_1$ and $s_2$ in the proof, respectively.
• Figure 5: A fine mixed subdivision and the corresponding graph.

Theorems & Definitions (29)

• Definition 2.1: San03, Definition 3.1
• Definition 2.2
• Theorem 2.3: ArdBil06, Theorem 6.2
• Definition 2.4
• Lemma 2.5
• proof
• Theorem 3.1
• Corollary 3.2
• Remark 3.3
• Definition 3.4: San03, Section 1.1