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We consider the uniform distribution on the set of partitions of integer n with c √ 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 → ∞ then the limit shape tends to known limit shape for unrestricted number of summands (see reference  ). If the growth is slower than √ n then the limit shape is universal (e −t ). We prove the invariance principle (central limit theorem for fluctuationsdoi:10.17323/1609-4514-2001-1-3-457-468 fatcat:kz22ek2dozaurdfue23fp47664