The Enumeration of Lattice Paths 3

We survey old and new results on the enumeration of lattice paths in the plane with a given number of turns, including the recent developments on the enumeration of nonintersecting lattice paths with a given number of turns. Motivations to consider such enumeration problems come from various elds, e.g. probability, statistics, combinatorics, and commutative algebra. We show that the appropriate tool for treating turn enumeration of lattice paths is the encoding of lattice paths in terms of two-rowed arrays. 3.1 Introduction In this article we consider lattice paths in the plane consisting of unit horizontal and vertical steps in the positive direction. We will be concerned with enumerating such lattice paths which have a given number of turns. By a turn, we mean a vertex of a path where the direction of the path changes. For example, (6; 4). Distinguishing between the two possible types of turns, we call a vertex of a path a NorthEast turn (NE-turn, for short) if it is the end point of a vertical step and at the same time the starting point of a horizontal step, and we call a vertex of a path an East-North turn (EN-turn, for short) if it is a point in a path P which is the end point of a horizontal step and at the same time the starting point of a vertical step.