The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model.

We solve both problems – namely the enumeration of planar Eulerian orientations and of 4-valent planar Eulerian orientations – by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, ${\mu }^n/{(n\log n)}^2$, prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order 2. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model.

Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices. In the 4-valent case, we also observe an unexpected connection with the enumeration of maps equipped with a spanning tree that is internally inactive in the sense of Tutte. This connection remains to be explained combinatorially.

至少自2000年以来，配备欧拉方向的平面图的枚举在组合物理学和理论物理学中都引起了人们的关注。4价图的情况特别有趣：这些方向是带有适当3色四边形的双射，而在物理学中它们对应于冰模型的配置。

我们通过将关联的生成函数表示为简单超几何级数的反函数（对于级数的组合）来解决两个问题，即平面欧拉方向和4价平面欧拉方向的枚举。使用这些表达式，我们推导出平面欧拉方向数的渐近行为，从而证明了科斯托夫，津恩·贾斯汀，埃尔维·普莱斯和古特曼的较早预测。这种行为${\mu }^{ñ}/{（ñ\mathrm{日志}ñ）}^2$防止关联的生成函数为D有限的。由于这些生成函数满足2阶非线性微分方程，因此它们仍然是微分代数的。最近，对于其他地图问题，尤其是配备有Potts模型的地图，证明了微分代数性。

我们的解决方案混合了递归和双射成分。特别是，初步的双射将我们的定向图转换成在其顶点上带有高度函数的图。在4价情况下，我们还观察到与配备了生成树的地图枚举（在Tutte的意义上是内部不活动的）的地图枚举之间的意外连接。这种联系还有待结合解释。