We study certain probability measures on partitions of n = 1,..., originated in representation theory, and demonstrate their connections with random matrix theory and multivariate hypergeometric functions. Our measures depend on three parameters including an analog of the β parameter in random matrix models. Under an appropriate limit transition as n → ∞, our measures converge to certain limit measures, which are of the same nature as one-dimensional log-gas with arbitrary β > 0. The first main result says that averages of products of "characteristic polynomials" with respect to the limit measures are given by the multivariate hypergeometric functions of type (2, 0). The second main result is a computation of the limit correlation functions for the even values of β.