Vertically constrained Motzkin-like paths inspired by bobbin lace

TL;DR

This paper introduces a new class of lattice paths inspired by bobbin lace, establishing bijections with Motzkin and Dyck paths, and deriving generating functions for their enumeration under vertical step constraints.

Contribution

It presents explicit bijections between vertically constrained paths and classical Motzkin, Dyck, Schr"{o}der, and Delannoy paths, extending combinatorial models inspired by lace.

Findings

Bijection between constrained paths and Motzkin paths.

Enumeration formulas via generating functions for these paths.

Extension of classical path sets with vertical step constraints.

Abstract

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $\phi$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ and a bisection of the paths generated by $\mathfrak{M}$ is presented. In a similar manner, paths with the step vectors $\mathfrak{B}=$$\{\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$, the union of Dyck step vectors and constrained vertical steps, are examined. We show, using the same $\phi$ mapping, that there is a bijection between vertically constrained $\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices. Generating functions are derived to enumerate these vertically constrained, partially directed paths when restricted to the half and quarter-plane. Finally, we extend Schr\"{o}der and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schr\"{o}der paths that are smooth (do not change direction) at a regular horizontal interval.