Bijections for Weyl Chamber walks ending on an axis, using arc diagrams and Schnyder woods

TL;DR

This paper introduces bijections connecting Weyl chamber walks, arc diagrams, and combinatorial objects like Young tableaux and Baxter permutations, using Schnyder woods to extend results across dimensions.

Contribution

It presents new bijections for Weyl chamber walks, standard Young tableaux, and Baxter permutations, employing arc diagrams and Schnyder woods, with extensions to higher dimensions.

Findings

New bijections for Weyl chamber walks and Young tableaux

A bijection between Baxter permutations and walks ending on an axis

Extension of bijective strategies to higher dimensions

Abstract

In the study of lattice walks there are several examples of enumerative equivalences which amount to a trade-off between domain and endpoint constraints. We present a family of such bijections for simple walks in Weyl chambers which use arc diagrams in a natural way. One consequence is a set of new bijections for standard Young tableaux of bounded height. A modification of the argument in two dimensions yields a bijection between Baxter permutations and walks ending on an axis, answering a recent question of Burrill et al. (2016). Some of our arguments (and related results) are proved using Schnyder woods. Our strategy for simple walks extends to any dimension and yields a new bijective connection between standard Young tableaux of height at most $2k$ and certain walks with prescribed endpoints in the $k$-dimensional Weyl chamber of type D.