The limit shape and fluctuations of random partitions of naturals with fixed number of summands

We consider the uniform distribution on the set of partitions of integer $n$ with $c\sqrt n$ numbers of summands, $c>0$ is a positive constant. We calculate the limit shape of such partitions, assuming $c$ is constant and $n$ tends to infinity. If $c\to\infty$ then the limit shape tends to known limit shape for unrestricted number of summands (see references). If the growth is slower than $\sqrt n$ then the limit shape is universal ($e^{-t}$). We prove the invariance principle (central limit theorem for fluctuations around the limit shape) and find precise expression for correlation functions. These results can be interpreted in terms of statistical physics of ideal gas, from this point of view the limit shape is a limit distribution of the energy of two dimensional ideal gas with respect to the energy of particles. The proof of the limit theorem uses partially inversed Fourier transformation of the characteristic function and refines the methods of the previous papers of authors (see references).