  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.
