  We present a Hamiltonian description of height-restricted forward-moving
(Dyck-like) paths on the plane, in which generating functions for the paths
arise as matrix elements of the Hamiltonian propagator. Using this approach we
evaluate the generating function for arbitrary starting and ending points of
the paths and express it as a ratio of determinants. Exploiting a previously
established connection between random walks and quantum exclusion statistics,
we express the generating function in terms of grand partition functions and
present an alternative simpler form for its logarithm that makes explicit its
polynomial structure.
