In the 1990 paper by Charles Knessl and Joseph Keller, the authors prove the asymptotic result (for $n>>1, k=O(1)$, your number is asymptotic to:
$\dfrac{n^{k-1}}{k[{k-1]!}^2}.$
They show a number of other related asymptotic results.
In the 1990 paper by Charles Knessl and Joseph Keller, the authors prove the asymptotic result (for $n>>1, k=O(1)$, your number is asymptotic to:
$\dfrac{n^{k-1}}{k[{k-1]!}^2}.$
They show a number of other related asymptotic results.