We consider a set of partitions of natural number n on distinct summands with uniform distribution. We investigate the limit shape of the typical partition as n --> infinity, which was found in [A. M. Vershik, Funct. Anal. Appl., 30 (1996), pp. 90-105], and fluctuations of partitions near its limit shape. The geometrical language we use allows us to reformulate the problem in terms of random step functions (Young diagrams). We prove statements of local limit theorem type which imply that joint distribution of fluctuations in a number of points is locally asymptotically normal. The proof essentially uses the notion of a large canonical ensemble of partitions.