Title: Coincidence? Perhaps.

Abstract:

In combinatorial enumeration it is often the case that two apparently different problems yield the same answer, or sequence of answers (powers of two, the Fibonacci sequence, and the Catalan sequence are particularly prevalent). Usually this is dismissed as “just coincidence” or as a consequence of simple formation rules for the objects being counted. However, it is less easy to understand the ubiquity of such coincidences in more complex settings. Recently, Mathilde Bouvel and I considered coincidences that occur in subsets of the “Catalan universe”. I will describe some simple and general conditions that are sufficient for coincidence (and are conjecturally necessary) which also explain why such coincidences are so common in this setting.