July 13, 10:00 ~ 11:00
Functional equations in enumerative combinatorics
CNRS, Université de Bordeaux, France - email@example.com
A basic idea in enumerative combinatorics is to translate the recursive structure of a class of objects into recurrence relations defining the sequence of numbers that counts them. In many cases, these recurrence relations translate further into functional equations, of various types, defining the associated generating functions.
Such equations give an answer to the counting problem, but it is not completely satisfactory if one cannot decide whether their solution belongs, by any chance, to a more classical family of functions: could it be rational, or algebraic? Could it satisfy a polynomial differential equation? Or even a linear one?
These are recurring questions in enumerative combinatorics, and they raise attractive problems at the intersection of algebra, analysis and computer algebra. We will present a few results in this direction, and illustrate them with many examples.