  We derive asymptotic formulas for the number of integer partitions with given
sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set
$J \subset \mathbb N$. The method we use is based on the `principle of maximum
entropy' of Jaynes. This principle leads to an intuitive variational formula
for the asymptotics of the logarithm of the number of constrained partitions as
the solution to a convex optimization problem over real-valued functions.
