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

We consider an integer partition λ 1 ⩾ ⋯ ⩾ λ ℓ , ℓ ⩾ 1 , chosen uniformly at random among all partitions of n such that λ 1 / λ 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 −1 λ ⌊ x m n ⌋ , x ∈ [ 0 , 1 ] ( m n = a n 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 Pakʹs 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 .