Further thoughts on ordered partitions

摘要

In [1] the authors consider the problem of enumerating the ways of writing n ∈ N as a sum of positive integers, where the order of the summands is important. So, for example, I + 1 + 2 and 1 + 2 + 1 would be regarded as distinct ways of expressing 4 as a sum of positive integers. This enumeration problem is equivalent to that of obtaining the cardinality of the set of finite sequences given by B_n= {(α_1, α_2, ... , α_k) : k, α_l, α_2, ... , a_k ∈ N; a_1 + a_2 + ... + a_k = n} . It is shown in [1] that |B_n|=2~(n-1).(1) However, the method used there to prove that this is the case was somewhat cumbersome, and indeed the purpose of this note is to demonstrate some rather more elegant ways of obtaining this result.