Publications and Preprints

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 $m \sim An^{\alpha}$ 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 Erd$\ddot{o}$s and Lehner~\cite{erdos} that concerns the relation between the number of integer partitions and compositions when $\alpha = \frac{1}{3}.$