论文标题
$ k $ - 平面树和$ k $的精制枚举
Refined enumeration of $k$-plane trees and $k$-noncrossing trees
论文作者
论文摘要
$ k $ - 平面树是一种平面树，其顶点的标签在$ 1 $和$ k $之间分配，以至于沿任何边缘的标签总和不超过$ k+1 $。已知这些树与$（k+1）$ - Ary树有关，它们用加泰罗尼亚数字的广义版本计数。我们证明了一个令人惊讶的简单精制计数公式，我们在其中用规定的每种标签数量计算树木。该公式衍生出几种推论，并以$ k $ noncrossing Trees的形式证明了类似的定理，这是一个类似定义的标记为非交叉树的家族，与$（2K+1）$ - Ary Trees相关。
A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to $(2k+1)$-ary trees.
