Catalan, Motzkin, and Riordan numbers Original Research Article

Catalan numbers Cn are widely known and studied and more recently the Motzkin numbers Mn have been celebrated. Closely joined to the Motzkin sequence is a sequence of unnamed numbers γn, also growing in importance. Here they are denoted by Rn, where R stands for Riordan. In 1977, using the planar coloring schemes defined by W. T. Tutte, I discovered that these numbers answer an old problem about linearly independent chromatic polynomials. Planar coloring schemes for an n-ring can be viewed as the subset of cyclically spaced noncrossing partitions of an n-cycle, and they define a natural geometric basis for chromatic polynomials on the n-ring. This article presents a unified overview of Catalan, Motzkin, and R-numbers, intended as a primer of sequences and techniques, combinatorial structure and recursion, generating function equations, difference triangles, and Lagrange inversion. Direct combinatorial correspondences are highlighted.