1-Wasserstein distance on the standard simplex

Wasserstein distances provide a metric on a space of probability measures. We consider the space $\Omega$ of all probability measures on the finite set $\chi = \{1, \dots ,n\}$ where $n$ is a positive integer. 1-Wasserstein distance, $W_1(\mu,\nu)$ is a function from $\Omega \times \Omega$ to $[0,\infty)$. This paper derives closed form expressions for the First and Second moment of $W_1$ on $\Omega \times \Omega$ assuming a uniform distribution on $\Omega \times \Omega$.