  The number of planar Eulerian maps with n edges is well-known to have a
simple expression. But what is the number of planar Eulerian orientations with
n edges? This problem appears to be difficult. To approach it, we define and
count families of subsets and supersets of planar Eulerian orientations,
indexed by an integer k, that converge to the set of all planar Eulerian
orientations as k increases. The generating functions of our subsets can be
characterized by systems of polynomial equations, and are thus algebraic. The
generating functions of our supersets are characterized by polynomial systems
involving divided differences, as often occurs in map enumeration. We prove
that these series are algebraic as well. We obtain in this way lower and upper
bounds on the growth rate of planar Eulerian orientations, which appears to be
around 12.5.
