Solving multivariate functional equations

TL;DR

This paper introduces a novel method for solving multivariate functional equations involving generating functions, enabling explicit formulas and practical coefficient calculations for complex combinatorial objects.

Contribution

The paper develops a new approach to solve multivariate functional equations, providing explicit formulas and applications to combinatorics and permutation problems.

Findings

Derived a formula for multivariate generating functions as sums over finite sequences

Applied the method to compute coefficients for parallelogram polyominoes

Solved a problem related to fully commutative affine permutations

Abstract

This paper presents a new method to solve functional equations of multivariate generating functions, such as $$F(r,s)=e(r,s)+xf(r,s)F(1,1)+xg(r,s)F(qr,1)+xh(r,s)F(qr,qs),$$ giving a formula for $F(r,s)$ in terms of a sum over finite sequences. We use this method to show how one would calculate the coefficients of the generating function for parallelogram polyominoes, which is impractical using other methods. We also apply this method to answer a question from fully commutative affine permutations.