Sets of dying period sets for word length n

1. Dying period sets are defined with respect to the genealogy of period sets, which is a tree that describes how period sets of length $n>2$ are algorithmically derived from their "parent" period set of length $n-1$. The algorithm is incremental, that is given the set of period sets for length $n-1$ it computes all period set of length $n$. In this genealogy, which is a tree, whose root is the unique period set of length $1$. Each node at depth $n$ in the tree represents a unique period set for length $n$. Each node has at most two children. Those nodes at depth $n$ that have no children are said to be dying at length $n$. 2. There exist period sets that never dies, meaning that the genealogy is infinite.1. 消亡周期集（dying period sets）的定义基于周期集谱系（genealogy of period sets），该谱系是一种树结构，用于描述长度$n>2$的周期集如何通过算法从其长度为$n-1$的“父周期集”衍生而来。该算法为增量式算法：给定长度为$n-1$的周期集集合，即可计算出所有长度为$n$的周期集。在该树状谱系中，根节点为唯一的长度为1的周期集；树中深度为$n$的每个节点对应一个唯一的长度为$n$的周期集，且每个节点至多拥有两个子节点。深度为$n$且无子节点的节点，被称为在长度$n$处消亡的周期集。 2. 存在永不消亡的周期集，这意味着该谱系是无限的。