Homework 1, Problem 4

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 n are there?” 



  1. 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 n, say n = 5. Now take a composition of 53+2. For each component in the composition (in this case, for 3 and for 2) take a composition. In my example, I’ll take … 1+2 and 1+1. You end up then with these two compositions (1+2,1+1) and thats one of the items you could have gotten. But you could have also taken the initial composition 2+2+1 and then taken the compositions (1+1,2,1) and that would be another possible item of the items you (we) are trying to count. Any thoughts? Does it even make sense?

  2. Dear Santiago, that’s exactly the way i counted it (vectors of vectors), and the result is something nice enough, so go ahead! 🙂

  3. Thanks! It’s clear to me now what it’s asking for.

  4. ale7bravo

    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:

    • 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 9, 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.

  5. Are these compositions of compositions the same: (1+1,2,1), (1,1+2,1)?

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