Generating function rationality for anisotropic vicious walk configurations on the directed square lattice

Guttmann and Vöge introduced a model of f-friendly walkers and argued that a generating function for the number of n-walker configurations making a total of k left steps is a rational function with denominator (1 - xⁿ)^{k + 1}. They also found that for f = 0, 1 and 2 the sums of the numerator coefficients for watermelon configurations in which each of 3 walkers made w left steps were 3-dimensional Catalan numbers. Here it is shown that for n vicious walker (f = 0) watermelon configurations the m^th coefficient of the numerator is the generalised Naryana number N(w, n, m) of Sulanke which is symmetric under interchange of w and n. The sums, C_{w, n}, of these coefficients as a sequence indexed by w are n-dimensional Catalan numbers or w-dimensional Catalan numbers if indexed by n. The unexpected symmetry in n and w is seen to follow from duality.

Inui and Katori introduced Fermi walk configurations which are non-crossing subsets of the directed random walks between opposite corners of a rectangular l × w grid. They related these to Bose configurations which biject to vicious walker watermelon configurations. Bose configurations include multisets. Here we consider generating functions for the numbers of configurations in which l and w are fixed. It is found that the maximum number of walks in a Fermi configuration is lw + 1 and the number of configurations corresponding to this number of walks is C_{l, w}. This limit on the number of walks in a Fermi configuration leads to the rationality of the Bose generating function and by duality to the rationality of the generating function of Guttmann and Vöge.