I think I’m confused about what this question is asking. Can someone rephrase it for me? I can’t tell if it’s a single question or two questions, where the first is simply, “how many compositions of 
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Enumerative Combinatorics 
I’m having my doubts about this question as well. But I think its just a single question, since, if there were two separate ones, then … the first one would be “how many compositions of n are?” but then what would the second question be?
I’m taking it like this. Take an
, say 
. Now take a composition of 
… 
. For each component in the composition (in this case, for 3 and for 2) take a composition. In my example, I’ll take … 
and 
. You end up then with these two compositions 
and thats one of the items you could have gotten. But you could have also taken the initial composition 
and then taken the compositions 
and that would be another possible item of the items you (we) are trying to count. Any thoughts? Does it even make sense?
What Santiago said is correct.
Dear Santiago, that’s exactly the way i counted it (vectors of vectors), and the result is something nice enough, so go ahead! 🙂
Thanks! It’s clear to me now what it’s asking for.
There is something I still don’t get. In the composition does the order matters? and, if I’m taking a composition, for example, of 5 does 5 counts as one? and so (5,0) is an item to count?
The answer to both of your questions is yes.
The order does matter and n is a composition of n.
For example:
n=3, then the compositions are:
1+1+1
1+2
2+1
3
Adding to what Juan Camilo said, if you are taking n=3, the whole set of thinks you want to count looks like this:
For 1+1+1: (1,1,1)
For 1+2 : (1,2),(1,1+1)
For 2+1 : (2,1),(1+1,1)
For 3 : (3), (1+2),(2+1),(1+1+1)
That means, take one composition. For each component of such composition, choose another composition. Do that for all compositions. If I am not wrong, the count for n=3 should be
, the number of “vectors” I was able to make. Notice that you will never get any zeros. So (3) counts, but (3,0) does not.
Are these compositions of compositions the same:
, 
?
Those two are considered to be different.