On the number of planar Eulerian orientations

TL;DR

This paper investigates the enumeration of planar Eulerian orientations, establishing algebraic generating functions for related subsets and supersets, and providing bounds on their growth rate.

Contribution

It introduces a novel approach using algebraic systems to bound the number of planar Eulerian orientations, a problem previously considered difficult.

Findings

Generated algebraic generating functions for subsets and supersets

Established bounds on the growth rate around 12.5

Provided a framework for approximating the enumeration of Eulerian orientations

Abstract

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.