Rook and queen paths with boundaries

TL;DR

This paper derives explicit algebraic formulas for counting Catalan rook and queen paths on a lattice, including extensions with spider steps and step-enumerators, advancing combinatorial path enumeration methods.

Contribution

It provides explicit algebraic generating functions for Catalan rook and queen paths, including new variants with spider steps and step-enumerator formulas.

Findings

Generating functions are algebraic and satisfy quadratic equations.

Explicit formulas are provided for paths ending at (n,n).

Extensions include paths with steeper 'spider' steps and step-enumerator versions.

Abstract

A rook path is a path on lattice points in the plane in which any proper horizontal step to the right or vertical step north is allowed. If, in addition, one allow bishop steps, that is, proper diagonal steps of slope 1, then one has queen paths. A rook or queen path is Catalan if it starts at the origin and stays strictly to the left of the line y = x-1. We give explicit formulas for the ordinary generating function of the number of Catalan rook and queen paths finishing at $(n,n).$ These generating functions are algebraic; indeed, they satisfy quadratic equations. In the second version, we also consider paths with "spider steps", that is, proper steps on lattice points with slope strictly greater than one. In the third version, we give step-enumerator versions of our results.