带边界曲面的单纯分解的计数
Enumerating simplicial decompositions of surfaces with boundaries
摘要：
众所周知，在其边界上具有$n+2$顶点的圆盘的三角剖分由$n$第卡塔兰数$C(n)=\frac{1}{n+1}{2n \choose n}$计数。本文将此问题推广到任意带边界的紧致曲面$S$。我们得到了曲面$S$的单分解的渐近数，其中$n$顶点在其边界上。更一般地，我们确定$S$的解剖的渐近数时，面对$δ$-$δ$属于一组容许度$Δ\subseteq \{3,4,5,...\}$。我们还给出了这种剖分的某些参数的极限定律。
英文摘要：
It is well-known that the triangulations of the disc with $n+2$ vertices on its boundary are counted by the $n$th Catalan number $C(n)=\frac{1}{n+1}{2n \choose n}$. This paper deals with the generalisation of this problem to any arbitrary compact surface $S$ with boundaries. We obtain the asymptotic number of simplicial decompositions of the surface $S$ with $n$ vertices on its boundary. More generally, we determine the asymptotic number of dissections of $S$ when the faces are $δ$-gons with $δ$ belonging to a set of admissible degrees $Δ\subseteq \{3,4,5,...\}$. We also give the limit laws of certain parameters of such dissections.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
