Unexam #3

Given in class on February 16, 2006.

I found that only one or two people had done the exercise that I had asked for during the last class. The result was that decided to assign this as the next unexam (this single problem only). I did not really want to do this because I had something else in mind for this unexam, but we are getting close to the end of class and here is a major topic in this course and it is not clear if the basics are well understood.

A partition of n is a representation of n as a sum of positive integers called summands or parts of the partition. The order of the summands is irrelevant.
pi_m(n) = the number of parititions of n into parts of size at most m.
p_m(n) = the number of partitions of n into at most m parts.

Q: How many integral solutions are there to the equation
x_1 + 2 x_2 + 3 x_3 + ... + m x_m = n with x_i=>0?

A: pi_m(n)

(Remark: Anna Yoon showed p_m(n) = pi_m(n), do not assume that this is obvious. It is a theorem in the book)

I am again looking for a solution which explains these answers clearly and completely.  I would like you to be direct and terse and I do not want to see irrelevant details in your solution.  Your solution should have everything necessary to explain this question and not more.  I am not looking for an explanation of 'how you solved this' or 'how you came to see the answer,' I am looking for a solution which you rewrote 3 or 4 times so that it is really clear and short.

Note: I am going to be much harsher when I grade this than I have on the past unexams because I think that you have some experience with what I am expecting. Please make sure that you follow the instructions.

On a 5 point scale here is what I am expecting:
5 points - a clear, short, direct explanation of why the number of solutions to the equation above is equal to the number of partitions of m into n parts (anything except this will not have a perfect score)
4 points - Not quite perfect
3 points - say something about both partitions of n and solutions to the equation and make a connection between the two but not why they are equivalent. Explanations which are long winded or unclear and include unnecssary statements. Manages to show that one quantity is less than or equal to the other (and maybe implies that the quantities are equal).
2 points - Explains with an example and implies that the general case is clear.
1 point - say something about either partitions or solutions but does not indicate why they are equal
0 points - makes no reference to either partitions or solutions to equations