A remark on certain restricted plane partitions and crystal melting model

In this paper, we provide formulas to calculate the partition functions of two types of plane partitions using the crystal melting model introduced by Okounkov, Reshetikhin and Vafa. As applications, we obtain a product formula for the partition function of the plane partitions with a limit boundary. A corollary of this formula is the demonstration of the equivalence between this partition function and the open 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 boundary along the $z$-axis direction.