2. Path Functions
Orient the lattice Z x Z so that the quadrant N x N is at the
lower right. Thus (i,j) is in row i and column j. A path P is a
finite sequence of horizontal or vertical jumps of length one. We
distinguish between two paths which cover the same route in different
orders. Let s(P) and e(P) denote the starting and ending points of
P, respectively. A sequence of paths P., P?, ... , P is connected if
e(P.) = s(P ^) for 1 j k _ m - 1 (and similarly for an infinite
sequence). In this case, P1 + P? + ... + P is the path formed by
following P.., P~ , ... , P in succession. Let -P be the path obtained
by traversing P backwards.
An elemental path consists of a single jump of length one either to
the right or downward. Every path P may be uniquely represented in the
form
(2.1) I «.P.,
i = 1 X X
where a. = ±1 and P. is an elemental path for 1 _ I _ m. Of course,
a ^ is just the it jump of P.
A path function is a mapping which associates Sp to the path P,
where Sp is a rational function of some underlying set of variables. If
P is an elemental path with s(P) = (i,j), we denote Sp by H. . or
V. . according to whether P is oriented horizontally or vertically,
1 J
respectively. It is tempting to view Sp as the result of integrating
over the path P. Unfortunately, we will not be able to distinguish
between the integrand and the variable of integration. Although the
analogy is incomplete, there is a striking set of similarities with the
process of integration.
We will define the path function Sp for elemental paths P and
extend to all paths by linearity
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