A Uniformly Distributed Statistic on a Class of Lattice Paths

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