Touchard's Drunkard

TL;DR

This paper presents a straightforward derivation based on Touchard's identity for counting walks that stay on the north side of the origin, contributing to combinatorial enumeration methods.

Contribution

It introduces a new derivation approach for enumerating constrained walks using Touchard's identity, simplifying previous methods.

Findings

Derived a simple enumeration formula for constrained walks

Connected Touchard's identity to walk enumeration

Provided insights into combinatorial walk constraints

Abstract

Based on Touchard's identity, a simple derivation is given for the enumeration of the N/S/E/W walks that remain on the north side of the origin.