Perfect partitions of a random set of integers

Abstract

Let $X_1,\dots, X_n$ be independent integers distributed uniformly on $\{1,\dots, M\}$, $M=M(n)\to\infty$ however slow. A partition $S$ of $[n]$ into $\nu$ non-empty subsets $S_1,\dots, S_{\nu}$ is called perfect, if all $\nu$ values $\sum_{j\in S_{\a}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $\nu$. For $\nu=2$, Borgs et al. proved, among other results, that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $\kappa:=\lim \tfrac{n}{\log M}>\tfrac{1}{\log 2}$, and that w.h.p. no perfect partition exists if $\kappa<\tfrac{1}{\log 2}$. We prove that w.h.p. no perfect partition exists if $\nu\ge 3$ and $\kappa<\tfrac{2}{\log \nu}$. We identify the range of $\kappa$ in which the expected number of perfect partitions is exponentially high. We show that for $\kappa> \tfrac{2(\nu-1)}{\log[(1-2\nu^{-2})^{-1}]}$ the total number of perfect partitions is exponentially high with probability $\gtrsim (1+\nu^2)^{-1}$.