Combinatorics of linearization coefficients

The linearization problem for a family of polynomials is to find the coefficients in the product of two polynomials when expressed as a linear combination of the polynomials. This is an old problem and its connection to combinatorics goes back 40 years.
We propose a new approach to the combinatorial interpretations of linearization coefficients of orthogonal polynomials. We use separation of variables. We illustrate our approach by applying it to determine the number of perfect matchings, derangements, and other weighted permutation problems.