  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.
