Agarwal在2003年给出了一个与正整数的无序分拆与有序分拆相关的恒等式.本文主要研究二类有序分拆,即对于给定的正整数k,分别研究正整数分拆成分部量不小于k和分部量不大于k的有序分拆.分别给出这二类有序分拆与无序分拆相关的一些恒等式.另外,还分别得到了这二类有序分拆数的递归式.利用这些递归式,我们又得到了一些恒等式,并且给出了正整数分拆成分部量不大于k有序分拆数与广义高阶菲波拉契数的关系式.

An identity between partitions and compositions was obtained by Agarwal in 2003.In this paper,we mainly study two types of composition:one is the compositions of n into parts greater than k and the other is the compositions of n into parts equal to 1,2,...,k,for a given integer k.By using Agarwai's method,we obtain some identities between the two types of composition and partitions,respectively.In addition,some recurrences,the relationship between the number of the first type compositions and the number of some subsets of set {1,2,...,n},and the relationship between the number of the second type compositions and general higher order Fibonacci number are obtained.