Counting Lattice Paths by Narayana Polynomials

Let $d(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ using the steps (0,1), (1,0), and (1,1). Let $e(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ with permitted steps from the step set ${\bf N} \times {\bf N} - \{(0,0)\}$, where ${\bf N}$ denotes the nonnegative integers. We give a bijective proof of the identity $e(n) = 2^{n-1} d(n)$ for $n \ge 1$. In giving perspective for our proof, we consider bijections between sets of lattice paths defined on various sets of permitted steps which yield path counts related to the Narayana polynomials.