给定度序列的标号树的计数
Counting Labelled Trees with Given Indegree Sequence
摘要：
对于顶点集$[n]:=\{1,2,..., n\}$上的标记树，如果满足$i<j$，则将每条边$ij$的方向定义为$i\to j$。$T$的入度序列可以看作是一个划分$λ\vdash n-1$。在计算射影空间中曲线的割线平面时，产生了具有给定入度序列的树的计数问题。最近Ethan Cotterill给出了$[n]$上对应于一个划分$λ$的入度序列的树的个数公式.本文给出了Cotterill猜想的两个证明：一个是基于归纳法的“半组合”证明，另一个是双射证明。
英文摘要：
For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $λ\vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $λ$. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial" based on induction, the other is a bijective proof.
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。
