number of ordered partitions of integer

Since $5$ is a smallish number, it is reasonable to try to list all of the ordered partitions, and then count. First maybe, lest we forget, write down the trivial partition $5$. Then write down $4+1$, $1+4$. Now list all the ordered partitions with $3$ as the biggest number. This is easy, $3+2$, $2+3$, $3+1+1$, $1+3+1$, $1+1+3$. Continue. After not too long, you will have a complete list.

It so happens that for this type of problem, there is a simple general formula, which one might guess by carefully finding the number of ordered partitions of $1$, of $2$, of $3$, of $4$. And there are good ways of proving that the general formula holds. Let us deal with the case $n=5$.

Put $5$ pennies in a row, leaving a little gap between consecutive pennies. There are $4$ interpenny gaps. CHOOSE any number of these gaps ($0$, $1$, $2$, $3$, or $4$) to put a grain of rice into. Any such choice gives rise to a unique ordered partition of $5$, and all of them arise in this way. For example, the trivial partition $5$ comes from using no grain. The partition $4+1$ comes from putting a grain of rice after the $4$th penny. And so on. So there are exactly as many ordered partitions of $5$ as there are ways of choosing a SUBSET of the set of gaps. But a set of $4$ elements has $2^4$ subsets.

Or else one could attack the problem by induction. For example, let $P(n)$ be the number of ordered partitions of $n$. Now look at $P(n+1)$. Ordered partitions of $n+1$ are of two types: (i) last element $1$ and (ii) last element bigger than $1$. You should be able to see that there are $P(n)$ ordered partitions of $n+1$ of each type, meaning that $P(n+1)=2P(n)$.

But after all this fancy stuff, I would like to urge that you get your hands dirty, that you list and count the ordered partitions of $n$ for $n=1$, $2$, $3$, $4$, $5$, maybe even $6$.

Counting in binary the groups of 1s or 0s form the partitions. Half are the same so there are 2^(n-1). As to be expected this gives the same results as the gaps method, but in a different order.

Groups

0000 4 
0001 3,1 
0010 2,1,1 
0011 2,2 
0100 1,1,2 
0101 1,1,1,1 
0110 1,2,1 
0111 1,3

Gaps

000 4
001 3,1
010 2,2
011 2,1,1
100 1,3
101 1,2,1
110 1,1,2
111 1,1,1,1

So $4+1$ is one example. $2+2+1$ is another

• What kinds of things add up to 5? (only numbers greater than or equal to 1 are used).

• What's the least number of numbers you can use? What's the greatest number?

• What if you rearrange the order of something you already have? Do you get something new (if you consider at as ordered)?

• Have you done it already for 1,2,3, and 4? You might be able to use those to help with 5.