A Uniformly Distributed Statistic on a Class of Lattice Paths

Let Gn 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 zero. Let Dn denote the set of paths in Gn with steps restricted to (1, 0), (0, 1), (1, 1), the so-called Delannoy paths. Stanley has shown that |Gn| = 2n−1|Dn| and Sulanke has given a bijective proof. Here we give a simple statistic on Gn that is uniformly distributed over the 2n−1 subsets of [n − 1] = {1, 2, . . . , n} and takes the value [n − 1] precisely on the Delannoy paths.