The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank

Abstract

Let $A=(a_{ij})$ be a nonnegative row-stochastic or column-stochastic $n\times n$ matrix of rank at most $k$, where $1\le k\le n$. Write $n=qk+s$, where $0\le s<k$. The author conjectured in 2018 that, for every permutation $\sigma$ of $\{1,\ldots,n\}$, $\prod_{i=1}^n a_{i,\sigma(i)}\le q^{-q(k-s)}(q+1)^{-(q+1)s}$. We prove this bound and classify all equality cases.