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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.

EDIT for $k \ll n,$ they have the asymptotic too painful to typeset, but you can find in http://dl.dropbox.com/u/5188175/2101859.pdf, equation (2.27)

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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.