What is the order of binary partition function when number of partitions is fixed?

Given a number $N$, the number of ways to write it as a sum of powers of $2$ is called the binary partition function of $N$ and is well studied. But if the number partitions are fixed, then how do you write the number $N$ as the sum of $k$ powers of $2$, that is, are there are any known upper and lower bounds for number of partitions?