Each connected component of a mapping $\{1,2,...,n\}\rightarrow\{1,2,...,n\}$ contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint -- that exactly one component exists -- are also examined. For instance, the mean length of the longest cycle is $(0.7824...)\sqrt n$ in general, but for the special case, it is $(0.7978...)\sqrt n$, a difference of less than $2\%$.
翻译：映射 $1, 2,...,...,n ⁇ n ⁇ rightrowr ⁇ 1,2,...,...n ⁇ n ⁇ $包含一个独特的周期。 通过延迟差分方程或逆拉帕特变换可以概率地研究其中最大的部分。 最长的这种周期同样包含两种方法: 我们找到一个长度的( 明显的新的) 密度公式。 限制的影响 -- -- 确切地说就有一个组成部分存在 -- -- 也得到了研究。 例如, 最长周期的平均长度一般是$( 0.7824...)\ sqrt n$, 但在特殊情况下, 差异小于$( 0. 7978...)\ sqrt n$, 差额小于$ $ 。
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