Abstract:
The (classical) kernel method has been folklore in combinatorics and related fields for a long time. There have been several extensions of the kernel method throughout the years, one of the most recent ones being the vectorial kernel method by Asinowski, Bacher, Banderier and Gittenberger. It can be used for deriving generating functions for lattice paths (with steps of length one) that avoid a fixed pattern, i.e. a certain sequence of consecutive steps.
In this talk we will see a generalization of the vectorial kernel method to lattice paths with longer steps. We will also have a look at some applications of this generalization which can be used to prove a conjecture by Callan about the asymptotic behaviour of the expected number of ascents in Schröder paths.