Fine Mixed Subdivisions of a Dilated Triangle

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.