Uncorrelatedness sets of discrete uniform distributions via Vandermonde-type determinants

Given random variables $X$ and $Y$ having finite moments of all orders, their uncorrelatedness set is defined as the set of all pairs $(j,k)\in{\mathbb N}^2,$ for which $X^j$ and $Y^k$ are uncorrelated. It is known that, broadly put, any subset of ${\mathbb N}^2$ can serve as an uncorrelatedness set. This claim ceases to be true for random variables with prescribed distributions, in which case the need arises so as to identify the admissible uncorrelatedness sets. This paper studies the uncorrelatedness sets for positive random variables uniformly distributed on three points. Some general features of these sets are derived. Two related Vandermonde-type determinants are examined and applied to describe uncorrelatedness sets in specific cases.