  Noncommutative invariant theory is a generalization of the classical
invariant theory of the action of $SL(2,\IC)$ on binary forms. The dimensions
of the spaces of invariant noncommutative polynomials coincide with the numbers
of certain noncrossing partitions. We give an elementary combinatorial
explanation of this fact by constructing a noncrossing basis of the homogeneous
components. Using the theory free stochastic measures this provides a
combinatorial proof of the Molien-Weyl formula in this setting.
