Algebraic Combinatorics Seminar - Julien Courtiel

Title: Solving Prellberg and Mortimer's conjecture - bijection(s)
between Motzkin paths and triangular walks

Abstract:

In these difficult times, what we need to feel better is some colorful and elegant bijections.

This talk introduces the work we did with Andrew Elvey-Price (Tours, France) and Irène Marcovici (Nancy, France). Together we answered an open question from Mortimer and Prellberg, asking for a bijection between a family of walks inside a bounded triangular domain (think about a large equilateral triangle subdivided in several smaller equilateral triangles) and the famous Motzkin paths, but which have bounded height.

The used techniques for the proof are quite elementary, and seem to be robust. Indeed, in addition to solving Mortimer and Prellberg's conjecture, our approach enabled us to find a new surprising bijection between 3D-walks constrained inside a pyramid and some 2D-walks in a squared grid.