论文标题
树木和周期
Trees and cycles
论文作者
论文摘要
令$ t $为$ n $顶点上的一棵树。我们可以将$ t $的边缘视为顶点集的换位；他们的产品（按任何顺序）是循环置换。所有可能的循环排列出现（每次），并且仅当树是恒星时才出现。在本文中，我们找到了已实现的周期的数量，并为其他树木获得了每个周期的实现数量的一些结果。我们还解决了产生给定周期的树木数量的反问题。在途中，我们遇到了一些熟悉的数字序列，包括欧拉和大惊小怪的数字。
Let $T$ be a tree on $n$ vertices. We can regard the edges of $T$ as transpositions of the vertex set; their product (in any order) is a cyclic permutation. All possible cyclic permutations arise (each exactly once) if and only if the tree is a star. In this paper we find the number of realised cycles, and obtain some results on the number of realisations of each cycle, for other trees. We also solve the inverse problem of the number of trees which give rise to a given cycle. On the way, we meet some familiar number sequences including the Euler and Fuss--Catalan numbers.
