Perfect partitions of a random set of integers

TL;DR

This paper investigates the existence and quantity of perfect partitions of random integer sets, establishing thresholds based on the ratio of set size to logarithm of the maximum element, extending previous results for two partitions to more.

Contribution

It generalizes the understanding of perfect partitions from two subsets to any number of subsets, providing probabilistic thresholds for their existence and abundance.

Findings

No perfect partitions exist with high probability for rac{n}{\, ext{log}\, M} < frac{2}{ ext{log}\, u} when rac{n}{\, ext{log}\, M} o ext{constant}

Expected number of perfect partitions is exponentially high in certain rac{n}{\, ext{log}\, M} ranges

Number of perfect partitions is exponentially high with positive probability for sufficiently large rac{n}{\, ext{log}\, M}

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 $\gtrsim (1+\nu^2)^{-1}$.