The sum of all width-one matrices

Abstract

A nonnegative integer matrix is said to be width-one if its nonzero entries lie along a path consisting of steps to the south and to the east. These matrices are important in optimal transport theory: the northwest corner algorithm, for example, takes supply and demand vectors and outputs a width-one matrix. The problem in this paper is to write down an explicit formula for the sum of all width-one matrices (with given dimensions $n \times n$ and given sum $d$ of the entries). We prove two strikingly different formulas. The first, a ${}_4 F_3$ hypergeometric series with unit argument, is obtained by applying the Robinson-Schensted-Knuth correspondence to the width-one matrices; the second is obtained via Stanley-Reisner theory. Computationally, our two formulas are complementary to each other: the first formula outperforms the second if $d$ is fixed and $n$ increases, while the second outperforms the first if $n$ is fixed and $d$ increases. We also show how our result yields a new non-recursive formula for the mean value of the discrete earth mover's distance (i.e., the solution to the transportation problem), whenever the cost matrix has the Monge property.