Limit shape of a random integer partition with a bounded max-to-min ratio of parts sizes

We consider an integer partition @l"1>=...>[email protected]"@?, @?>=1, chosen uniformly at random among all partitions of n such that @l"1/@l"@? does not exceed a given number k>1. For k=2, Igor Pak had conjectured existence of a constant a such that the random function m"n^-^[email protected]"@?"x"m"""n"@?, [email protected]?[0,1] (m"n=an^1^/^2), converges in probability to y=f(x)>=1, f(0)=2, f(1)=1, whose graph is symmetric with respect to y=x+1. We confirm a natural extension of Pak's conjecture for k>1, and show that the limit shape y=f(x) is given by w^x^+^1+w^y=1, where w^k+w=1. In particular, for k=2, w is the golden ratio (5-1)/2.