Planar Maps and Continued Fractions
Planar Maps and Continued Fractions
复制标题
平面映射和连分数
DOI：
10.1007/s00220-011-1401-z
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发表时间：
2010
影响因子：
2.4
通讯作者：
E. Guitter
中科院分区：
文献类型：
--
作者：
J. Bouttier;E. Guitter
我们提出了两个地图枚举问题之间的意外联系。第一个包括计算具有规定长度边界的平面地图。第二个包括计算具有规定距离的两个点的平面地图。我们表明，在具有受控面度的一般地图类中，这两个问题的解决方案实际上分别通过其幂级数展开和连分数展开编码为相同的量。然后，我们使用解决第一个问题的已知技术来解决第二个问题。这种新颖的观点为计算一般平面地图的所谓距离相关两点函数提供了一种建设性的方法。我们证明并扩展了一些先前预测的精确公式，我们根据特定的 Schur 函数来识别这些公式。
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a constructive approach for computing the so-called distance-dependent two-point function of general planar maps. We prove and extend some previously predicted exact formulas, which we identify in terms of particular Schur functions.