  Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of
an integer $n$ into parts from the set $A$. The aim of this paper is to prove
several result concerning the existence of integer solutions of Diophantine
equations of the form $P_{A}(x)=P_{B}(y)$, where $A, B$ are certain finite
sets.
