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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