有序与无序之间的几个分拆恒等式和递归式

Some Identities and Recurrences between Compositions and Partitions

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.随后,产生了一系列与正整数的无序分拆与有序分拆相关的恒等式.在此基础上研究了2类新的与正整数的无序和有序之间的分拆恒等式,得到了正整数的"奇-偶-奇"有序分拆和最大分部量为偶数,其余分部量为奇数的无序分拆之间的恒等式以及正整数的"偶-奇-偶"有序分拆和"奇-偶"无序分拆之间的恒等式,并给出了它们的组合证明,同时给出了这2类有序分拆数的递归式.

An identity between partitions and compositions was obtained by Agarwal in 2003.Subsequently,other identities between partitions and compositions were obtained.In this paper,two new kinds of identities between partitions and compositions were studied.By using combination method,the identical relation between 'odd-even-odd' composition and the partition in which the largest part is an even and the other parts are all odd numbers are obtained and the identical relation between 'even-odd-even' composition and 'odd-even' partition are obtained.Furthermore,the recurrences between two types of compositions are given.