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{\bf Robert A. Sulanke}
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{\bf Counting  Lattice Paths by Narayana Polynomials}
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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.


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