0
$\begingroup$

I'm supposed to find the generating function of the number of partitions of a natural number n, where no summand can be higher than 3. For example, where n=8, the sum 2+3+3 is fine, but 2+2+4 isn't.

I was thinking of the following of using the following function: (1+x^1+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+x^9+...). The idea is I'm choosing how many "ones" to add to the sum with the first term ,how many "twos" with the second term and how many "threes" with the third. So if I wanted to say for n=8, and the sum 1+2+2+3 then I'll take x^1 from the first term, x^4 from the second and x^3 from the third.

I hope it is clear enough to understand. I don't have any final solutions to compare to, so I am wondering if this is at all correct.

Thanks for the help.

1 Answers 1

2

Yes, you are is correct, the generating function of the number of partitions of $n$ in which the greatest part is less than or equal to 3 is $$(1+x+x^2+\dots)(1+x^2+x^4+\dots)(1+x^3+x^6+\dots)=\frac{1}{(1-x)(1-x^2)(1-x^3)}$$ See OEIS sequence A001399.

  • 0
    Thanks! I have a follow-up question. It's the same thing only this time I'm looking for the number of partitions where there are excatly 3 summands. so If n=8 then 1+1+6 and 2+3+3 are fine, but 1+2+2+3 isn't. from your link it seems that this is the same answer, but I can't see why... – 2017-01-14
  • 0
    Take a look to the Ferrers Diagram (http://mathworld.wolfram.com/FerrersDiagram.html). Then trasform the rows in columns. For example 6+1+1 becomes 3+1+1+1+1. This is bijection between the partitions with exactly 3 summonds and the ones where the greatest part is less than or equal to 3. – 2017-01-14
  • 0
    I didn't really understand their explanation. Im not familiar yet with anything that have to do with graph theory or with ferrers diagram. can you expand? – 2017-01-14
  • 0
    Read this answer to a similar question: http://math.stackexchange.com/questions/389332/how-to-prove-using-ferrers-diagrams Is it clear now? – 2017-01-14
  • 0
    I think I understand how this explains why every natural number has the same number of partitions of exactly 3 summands and of summands no higher then 3 (they are mirror images of each other) but I'm supposed to reach an expression for the generating function of this problem without basing my answer on the fact that they are the same (mostly, since I'm then asked to show that they are the same since they have the same generating function...) – 2017-01-15