几个关于1-2有序分拆的恒等式及组合证明

Several identities and combinatorial proofs for 1-2 compositions

研究了正整数的两类1-2有序分拆,其中一类是正整数的首、末两端分部量都是1的1-2有序分拆,另一类是正整数的首、末两端分部量至少有一个是2的1-2有序分拆.首先得到了这些有序分拆数与Fibonacci数之间的一些关系式.进而,利用熟知的与Fibonacci数相关的有序分拆恒等式得到了这两类正整数的有序分拆数与分部量是奇数、分部量大于1、分部量是1或者2的有序分拆数之间的一些新的有序分拆恒等式,并给出了这些恒等式的组合双射证明.

Two classes of 1-2 compositions of positive integer are studied.One of them is the 1-2 compositions with parts of size 1 at the left and the right of positive integers,and the other is the 1-2 compositions with parts of size 2 at the left or the right of positive integers.Firstly,some relations between the number of these compositions and the Fibonacci numbers are obtained.And then using the well-known composition identities related to the Fibonacci numbers,several new composition identities between the number of these two classes of the compositions and the number of the compositions with parts of odd,the number of the compositions with parts of size greater than 1 and the number of the compositions with parts of size 1 or 2 are got.In addition,combinatorial bijective proofs of these identities are given.