  We study quantum integrable models with GL(3) trigonometric R-matrix solvable
by the nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such
models can be expressed in terms of a bilinear combination of the highest
coefficients. We show that in the models with GL(3) trigonometric R-matrix
there exist two different highest coefficients. We obtain various
representations for them in terms of sums over partitions. We also prove
several important properties of the highest coefficients, which are necessary
for the evaluation of the scalar products.
