论文标题
具有交替边界的平面星座的枚举
Enumeration of planar constellations with an alternating boundary
论文作者
论文摘要
具有边界的平面超图定义为具有边界的平面图，并具有适当的内在面部的双色。如果事件的颜色沿其轮廓交替，则据说边界交替。在本文中，我们考虑了按照每种颜色的内部的周长和程度分布来计数平面超图与交替边界的问题。该问题被转化为具有催化变量确定相应生成函数的功能方程。在星座的情况下 - 给定颜色的所有内在面都有$ m \ geq 2 $，并且其所有其他内在面的倍数为$ m $ - 我们完全求解了功能方程，并表明生成函数是代数并接受明确的官能参数。我们最终专注于欧拉三角剖分的情况 - 超图的所有内在面孔的学位$ 3 $ - 并计算第二作者在另一项作品中需要的渐近学，以证明已恢复后的平面欧拉三角形的融合到Brownian Map。
A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of innerfaces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function. In the case of constellations - hypermaps whose all inner faces of a given color have degree $m\geq 2$, and whose all other inner faces have a degree multiple of $m$ - we completely solve the functional equation, and show that the generating function is algebraic and admits an explicit rational parametrization. We finally specialize to the case of Eulerian triangulations - hypermaps whose all inner faces have degree $3$ - and compute asymptotics which are needed in another work by the second author, to prove the convergence of rescaled planar Eulerian triangulations to the Brownian map.
