集[1,n]上几个特殊映射的个数

Number of Several Special Mapping on [1,n] Set

本文首先给出集[1,n]上k-次幂等映射的概念,用置换群的循环分解给出集[1,n]上k-次幂等映射的计数公式,接着定义了连象映射和保小映射,不但得到它们的计数公式,还得到一个第2类斯特林数的关系式及保小映射总数指母函数系数的变化趋势,最后给出交错映射及上下置换的概念并给出关于上下置换指母函数简明表达式的一个简单证明.

The k - th idempotent mapping on the set [ 1 ,n] is firstly defined in this paper, the counting tormma of which is obtained by circulating decomposition of permutation groups. The definitions of small preserving mapping and connected image mapping are given afterwards together with their counting formula, a relational formula of the second stirling numbers and the trend of exponential generating function coefficient of the total amount of small preserving mapping. The concepts of alternative mapping as well as top and bottom permutation are also made clear. The concise expression of exponential generating function of top and bottom permutation is finally proved simply.