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Guttmann and Voge 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 of3 walkers made w left steps were3-dimensional Catalan numbers. Here it is shown that for n vicious walker (f =0 )watermelon configurations the mth 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, Cw,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  × 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 whichand w are fixed. It is found that the maximum number of walks in a Fermi configuration isw +1and the number of configurations corresponding to this number of walks is C,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 V¨ oge.

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