Combinatorics of orthogonal polynomials on the unit circle

Orthogonal polynomials on the unit circle (OPUC for short) are a family of polynomials whose orthogonality is given by integration over the unit circle in the complex plane. There are combinatorial studies on the moments of various types of orthogonal polynomials, including standard orthogonal polynomials, Laurent biorthogonal polynomials, and orthogonal polynomials of type \( R_I \). In this paper, we study the moments of OPUC from a combinatorial perspective. We provide three path interpretations for them: \L{}ukasiewicz paths, gentle Motzkin paths, and Schr\"oder paths. Additionally, using these combinatorial interpretations, we derive explicit formulas for the generalized moments of some examples of OPUC, including the circular Jacobi polynomials and the Rogers--Szeg\H{o} polynomials. Furthermore, we introduce several kinds of generalized linearization coefficients and give combinatorial interpretations for them.