Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein  and Brubaker, Bump and Friedberg  provided n is sufficiently large; their coefficients involve n-th order Gauss sums. The case where n is small is harder, and is addressed in this paper when Φ = Ar. "Twisted" Dirichet series are considered, which contain the series of  as a special case. These series are not Euler products, but due to the twisted multiplicativity of their coefficients, they are determined by their p-parts. The p-part is given as a sum of products of Gauss sums, parametrized by strict Gelfand-Tsetlin patterns. It is conjectured that these multiple Dirichlet series are Whittaker coefficients of Eisenstein series on the n-fold metaplectic cover of GLr+1, and this is proved if r = 2 or n = 1. The equivalence of our definition with that of Chinta  when n = 2 and r ≤ 5 is also established.