Multiplicity of summands in the random partitions of an integer

Abstract

In this paper, we prove a conjecture of Yakubovich regarding limit shapes of ‘slices’ of two-dimensional (2D) integer partitions and compositions of n when the number of summands m ~Anα for some A > 0 and \(\alpha < \frac{1}{2}\). We prove that the probability that there is a summand of multiplicity j in any randomly chosen partition or composition of an integer n goes to zero asymptotically with n provided j is larger than a critical value. As a corollary, we strengthen a result due to Erdos and Lehner (Duke Math. J.8 (1941) 335–345) that concerns the relation between the number of integer partitions and compositions when \(\alpha = \frac{1}{3}\).