Bijections for Weyl Chamber walks ending on an axis, using arc diagrams
and Schnyder woods
release_42q5jocf7rahfjgoc45llcg654
by
Julien Courtiel, Éric Fusy, Mathias Lepoutre, Marni Mishna
2017
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.
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