Expected value of the one-dimensional Earth Mover's Distance

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 [n] = {1,...,n} 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 [n]. 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 one-dimensional case. The EMD is then used in clustering analysis for a specific data set.