The height of watermelons with wall Extended

The model of vicious walkers was introduced by Fisher [4]. He gave a number of applications in physics, such as modelling wetting and melting processes. In general, the model of vicious walkers is concerned with p random walkers on a d-dimensional lattice. In the lock step model, at each time step all of the walkers move one step in any of the allowed directions, such that at no time any two random walkers share the same lattice point. A configuration that attracted much interest amongst mathematical physicists and combinatorialists is the watermelon configuration, which is a special case of the two dimensional vicious walker model. See Figure 1 for an example of a watermelon, where, for the moment, the broken line labelled 13 should be ignored. This configuration can be studied with or without presence of an impenetrable wall, and with or without deviation. We proceed with a description of p-watermelons of length 2n with wall (without deviation), which is the model underlying this paper. Consider the lattice in R2 spanned by the two vectors (1,1) and (1,−1). At time zero the walkers are located at the points (0,0),(0,2), . . . ,(0,2p− 2). The allowed directions for the walkers are given by the vectors (1,1) and (1,−1). Further, the horizontal line y = 0 is an impenetrable wall, that is, no walker is allowed to cross the x-axis. The walkers may now simultaneously move one step in one of the allowed directions, but such that at no time two walkers share the same place. Additionally we demand that after 2n steps all walkers are located at (2n,0),(2n,2), . . . ,(2n + 2p−2). Tracing the paths of the vicious walkers through the lattice we obtain a set of non-intersecting lattice paths with steps in the set {(1,1),(1,−1)}. In the case of watermelons without deviation, the i-th lattice