Hamiltonian and exclusion statistics approach to discrete forward-moving paths

TL;DR

This paper develops a Hamiltonian-based method to analyze height-restricted Dyck paths, connecting combinatorics with quantum exclusion statistics, and provides explicit generating functions for path length and area.

Contribution

It introduces a novel Hamiltonian and exclusion statistics framework to evaluate generating functions for Dyck paths with arbitrary endpoints.

Findings

Expressed generating functions as rational combinations of determinants.

Connected path enumeration with quantum exclusion statistics.

Provided explicit polynomial structure for the generating functions.

Abstract

We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an alternative, simpler form for its logarithm that makes its polynomial structure explicit.