具有指定度和围长的映射的统一双射
Unified bijections for maps with prescribed degrees and girth
摘要：
本文给出了平面地图的统一双射构造，并控制了面度和围长。回想一下，围长是最小圈的长度，因此围长至少为$d=1,2,3$的映射分别是一般映射、无环映射和简单映射。对于每一个正整数$d$，我们得到了围长为$d$且根面为$d$的平面映射类的双射.然后，我们得到更一般的双射结构的环形地图（地图与两个区别根面）的围长至少$d$。我们的双射关联到每个地图的装饰平面树，和非根面的程度$k$的地图对应的顶点的程度$k$的树。作为特殊情况下，我们恢复了几个已知的双射二分地图，无环三角剖分，简单的三角剖分，简单的四边形剖分等，我们的工作统一，大大扩展了这些双射的建设。在计数方面，我们得到了每一个整数$d$的周长$d$的平面映射的生成函数$F_d(x_d,x_{d+1},x_{d+2},...)$的表达式$d$度的根面，其中变量$x_k$计数度$k$的非根面。Bouttier、Di Francesco和Guitter已经双射地获得了$F_1$的表达式，但对于$d\geq 2$，$F_d$的表达式是新的。我们还得到了一个表达式的生成函数$\G_{p,q}^{(d,e)}(x_d,x_{d+1},...)$的环形地图的根面度$p$和$q$，使两个根面之间的循环的长度至少$e$，而其他的循环的长度至少$d$。我们的策略是将所有双射作为作者在前一篇文章中介绍的单个“主双射”的特殊化。为了使用这种方法，我们表现出一定的“规范方向”的特征与规定的围长约束的地图。
英文摘要：
This article presents unified bijective constructions for planar maps, with control on the face degrees and on the girth. Recall that the girth is the length of the smallest cycle, so that maps of girth at least $d=1,2,3$ are respectively the general, loopless, and simple maps. For each positive integer $d$, we obtain a bijection for the class of plane maps (maps with one distinguished root-face) of girth $d$ having a root-face of degree $d$. We then obtain more general bijective constructions for annular maps (maps with two distinguished root-faces) of girth at least $d$. Our bijections associate to each map a decorated plane tree, and non-root faces of degree $k$ of the map correspond to vertices of degree $k$ of the tree. As special cases we recover several known bijections for bipartite maps, loopless triangulations, simple triangulations, simple quadrangulations, etc. Our work unifies and greatly extends these bijective constructions. In terms of counting, we obtain for each integer $d$ an expression for the generating function $F_d(x_d,x_{d+1},x_{d+2},...)$ of plane maps of girth $d$ with root-face of degree $d$, where the variable $x_k$ counts the non-root faces of degree $k$. The expression for $F_1$ was already obtained bijectively by Bouttier, Di Francesco and Guitter, but for $d\geq 2$ the expression of $F_d$ is new. We also obtain an expression for the generating function $\G_{p,q}^{(d,e)}(x_d,x_{d+1},...)$ of annular maps with root-faces of degrees $p$ and $q$, such that cycles separating the two root-faces have length at least $e$ while other cycles have length at least $d$. Our strategy is to obtain all the bijections as specializations of a single "master bijection" introduced by the authors in a previous article. In order to use this approach, we exhibit certain "canonical orientations" characterizing maps with prescribed girth constraints.
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