  We give a combinatorial interpretation of vector continued fractions obtained
by applying the Jacobi-Perron algorithm to a vector of $p\geq 1$ resolvent
functions of a banded Hessenberg operator of order $p+1$. The interpretation
consists in the identification of the coefficients in the power series
expansion of the resolvent functions as weight polynomials associated with
Lukasiewicz lattice paths in the upper half-plane. In the scalar case $p=1$
this reduces to the relation established by P. Flajolet and G. Viennot between
Jacobi-Stieltjes continued fractions, their power series expansion, and Motzkin
paths. We consider three classes of lattice paths, namely the Lukasiewicz paths
in the upper half-plane, their symmetric images in the lower half-plane, and a
third class of unrestricted lattice paths which are allowed to cross the
$x$-axis. We establish a relation between the three families of paths by means
of a relation between the associated generating power series. We also discuss
the subcollection of Lukasiewicz paths formed by the partial $p$-Dyck paths,
whose weight polynomials are known in the literature as genetic sums or
generalized Stieltjes-Rogers polynomials, and express certain moments of
bi-diagonal Hessenberg operators.
