I will define an m-regular non-crossing partition and prove a bijection between certain m-regular non-crossing partitions of n and certain (m-1)-regular non-crossing partitions of n − 1. This bijection allows us to express the Narayana numbers (which count the number of plane trees with n + 1 vertices and k leaves) in terms of the Catalan numbers (which count any of 166 things, a la Stanley).
This talk is based on a paper by Chen, Deng, and Du.