Summary: A distribution whose normalization constant is an \(A\)-hypergeometric polynomial is called an \(A\)-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an \(A\)-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) \(A\)-hypergeometric distributions. Then, the maximum likelihood estimation of the \(A\)-hypergeometric distribution associated with the rational normal curve, which is an algebraic exponential family, is discussed. The information geometry of the Newton polytope is useful for analyzing the full and the curved exponential family. Algebraic methods are provided for evaluating the \(A\)-hypergeometric polynomials.