A uniformly distributed parameter on a class of lattice paths

Let G_n denote the set of lattice paths from (0,0) to (n,n) with steps of the form (i,j) where i and j are nonnegative integers, not both 0. Let D_n denote the set of paths in G_n with steps restricted to (1,0), (0,1), (1,1), so-called Delannoy paths. Stanley has shown that | G_n | = 2^(n-1) | D_n | and Sulanke has given a bijective proof. Here we give a simple parameter on G_n that is uniformly distributed over the 2^(n-1) subsets of [n-1] = {1,2,...,n-1} and takes the value [n-1] precisely on the Delannoy paths.