From a combinatorial point of view, we consider the earth mover’s
distance (EMD) associated with a metric measure space. The specific
case considered is deceptively simple: Let the finite set of integers
$\left[n\right]=\left\{1,\dots ,n\right\}$ be
regarded as a metric space by restricting the usual Euclidean distance on the real
numbers. The EMD is defined on ordered pairs of probability distributions on
$\left[n\right]$. We
provide an easy method to compute a generating function encoding the values of
EMD in its coefficients, which is related to the Segre embedding from projective
algebraic geometry. As an application we use the generating function to compute the
expected value of EMD in this onedimensional case. The EMD is then used in
clustering analysis for a specific data set.
