Compositional Inversion and Appell Sequences is a brief set of notes (pdf) on the relation between a Lagrange compositional inversion of a function in terms of the coefficients of the reciprocal of the function and a general set of Appell polynomials (special Scheffer sequences). The partition polynomials for the inversion form an Appell sequence w.r.t. the coefficient of the linear degree of the reciprocal and have coefficients related to a refinement of the Narayana numbers (A134264 / A001263) and enumeration of Dyck (A125181) and Lukasiewicz paths, non-crossing partitions, and certain types of trees.
For a general discussion of non-crossing partitions and their intersection with other diverse domains of math, see “Generalized noncrossing partitions and combinatorics of Coxeter groups” by Drew Armstrong with extensive references.
For visual presentations of some combinatorics underlying many of the integer arrays here, see http://www.robertdickau.com/default.html#math.
At Play in an Orchard: Catalan, Riordan Appels and Lagrange Pairs reiterates the above notes and extends them to show how the Catalan numbers (OEIS-A000108) are related to other special number sequences, such as the Fibonacci (A000045), Motzkin sums / Riordan (A005043), and Fine (A000957), through simple compositions, and through simple interpolation to associated polynomials, such as the Fibonacci polynomials (A030528) and those related to the Dyck / Lukasiewicz lattice paths, non-crossing partitions, and trees of A091867. In addition, it provides a procedure for constructing or identifying families related to other parent sequences.