Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The number of planar Eulerian maps with n edges is ll-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,  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.  prove that these series are algebraic as ll.  obtain in this way lor and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5.
['Nicolas Bonichon', 'Mireille Bousquet-Mélou', 'Paul Dorbec', 'Claire Pennarun']
I am writing a research paper on a idea On the number of planar Eulerian orientations
[{"from": "human", "value": "I am writing a research paper on a idea On the number of planar Eulerian orientations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The number of planar Eulerian maps with n edges is ll-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,  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.  prove that these series are algebraic as ll.  obtain in this way lor and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5."}]