论文标题
在准多项式上计数平面紧密地图
On quasi-polynomials counting planar tight maps
论文作者
论文摘要
紧密地图是一张标有某些顶点的地图，因此标记了$ 1 $的每个顶点。我们给出一个数字的明确公式$ n_ {0，n}（d_1，\ ldots，d_n）$的平面紧密地图，带有$ n $标记的处方学位的标记面孔$ d_1，\ ldots，d_n $，在标有标记的顶点的facex face face face a face a face a face $ 0 $ $ 0 $ $ $ $。如前所述，它是$（d_1，\ ldots，d_n）$中的准级别元素。我们的派生是培养的，是基于平面图的切片分解。在非双方情况下，我们还依靠两种森林的枚举结果。我们讨论与非必然紧张地图的枚举的联系。特别是，我们将Tutte的经典切片公式概括为所有非两部分地图。
A tight map is a map with some of its vertices marked, such that every vertex of degree $1$ is marked. We give an explicit formula for the number $N_{0,n}(d_1,\ldots,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees $d_1,\ldots,d_n$, where a marked vertex is seen as a face of degree $0$. It is a quasi-polynomial in $(d_1,\ldots,d_n)$, as shown previously by Norbury. Our derivation is bijective and based on the slice decomposition of planar maps. In the non-bipartite case, we also rely on enumeration results for two-type forests. We discuss the connection with the enumeration of non necessarily tight maps. In particular, we provide a generalization of Tutte's classical slicings formula to all non-bipartite maps.
