  In this paper, we provide formulas calculating the partition functions of two
types of plane partitions using the crystal melting model method introduced by
Okounkov, Reshetikhin and Vafa. As applications, we obtain a product formula
for the partition function of the plane partitions with a limit shape boundary.
A corollary of this formula is the demonstration of the equivalence between
this partition function and the open-closed string amplitude of the
double$-\mathbb{P}^1$ model. We also derive a product formula for the partition
function of symmetric plane partitions with a limit shape boundary.
