  We study random partitions $\lambda=(\lambda_1,\lambda_2,...,\lambda_d)$ of
$n$ whose length is not bigger than a fixed number $d$. Suppose a random
partition $\lambda$ is distributed according to the Jack measure, which is a
deformation of the Plancherel measure with a positive parameter $\alpha>0$. We
prove that for all $\alpha>0$, in the limit as $n \to \infty$, the joint
distribution of scaled $\lambda_1,..., \lambda_d$ converges to the joint
distribution of some random variables from a traceless Gaussian
$\beta$-ensemble with $\beta=2/\alpha$. We also give a short proof of Regev's
asymptotic theorem for the sum of $\beta$-powers of $f^\lambda$, the number of
standard tableaux of shape $\lambda$.
