Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

Abstract

This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0°, 60°, 120°, 180°, 240°, 300°) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height.
 We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a 2n-to-one function from generic triangular walks to triangular walks only using directions 0°, 120°, 240°. The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections.
 Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.

Affine Gordon-Bender-Knuth identities for cylindric Schur functions

“…In this section we consider the special case h = 1 of Corollary 7.8. We show that if h = 1 then the first two identities (7.10) and (7.11) are equivalent to a result of Mortimer and Prellberg [23] whose bijective proof has been given by Courtiel, Elvey Price, and Marcovici [5]. We then give a bijective proof of the last two identities (7.12) and (7.13) for h = 1.…”

“…Let p 0 = (0, 0) and for i ∈ [n] define The identities (8.1) and (8.2) were first proved by Mortimer and Prellberg [23] using the kernel method. Recently, Courtiel, Elvey Price, and Marcovici [5] found a bijective proof of these two identities. For the rest of this section we give a bijective proof of (8.3) and (8.4).…”

“…. For example, the (2, 3)-cylindric row-strict tableau in the right diagram of Figure 3 corresponds to the lattice path (0, 0) → (1, 1) → (2, 2) → (3, 2) → (4, 3) → (5,4).…”

“…, 2n, we now read the vector of x-coordinates of the points of the lattice paths that are at height j. For our running example from Figure 6, we get the sequence (1,2,3,4), (1,2,4,5), (1,2,3,5), (2,3,4,6), (1,3,4,6), (1,4,5,7), (…”