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

We consider an integer partition λ1⩾⋯⩾λℓ, ℓ⩾1, chosen uniformly at random among all partitions of n such that λ1/λℓ 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 , x∈[0,1] (mn=an1/2), 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 wx+1+wy=1, where wk+w=1. In particular, for k=2, w is the golden ratio .