Method for Obtaining Coefficients of Powers of Multivariate Generating Functions

There are several general concepts that allow obtaining explicit formulas for the coefficients of generating functions in one variable by using their powers. One such concept is application compositae functions. In previous studies, we have introduced a generalization multivariate and defined basic operations on bivariate The use these helps to obtain two variables. this paper, expand case three addition, describe way applying rational To confirm effectiveness proposed method, present detailed examples its function related Aztec diamond permutations with cycles.