| Abstract: When we cannot solve an enumeration problem, we may wonder whether we are just not using the right methods, or the problem is actually difficult. If it is actually difficult, how do we measure its level of difficulty? In this talk, we will survey recent methods to prove that the generating function of a combinatorially defined sequence is not rational or not algebraic in situation when we know very little about that generating function.
Recently, these methods have been successfully applied in the area of pattern avoiding permutations, but they are applicable in other settings as well. While we will use techniques from analytic combinatorics, the talk will be self-contained and accessible for a general mathematics audience. |