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How do I find the amount of possible ordered partition of $n$, given set of positive integer $S$? Here's an example,

With $n = 4$ and $S = \{1, 3, 4\}$, we should have $4$, as $(1,1,1,1)$, $(1, 3)$, $(3, 1)$, and $(4)$ are the solutions.

I've seen this page but didn't have a clue how he transformed the equation to functions of $x$. Neither do I know the name of the transformation technique nor what is $x$.

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math_sucker
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1 Answers1

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In your example, you're looking for the coefficient of $x^4$ in the generating function $1/(1-x-x^3-x^4) $. The general case is similar. In particular, in this way you can obtain asymptotics as well as an explicit formula for the number of representations of $n$ as an ordered partition with parts $1,3,4$. The idea is to decompose the generating function into "partial fractions".

I believe that all roots of $1-x-x^3-x^4$ are distinct, and so we can write $$\frac{1}{1-x-x^3-x^4}=\sum_{i=1}^{4} \frac{b_i}{1-a_ix}, $$ from which we can read the formula $\sum_{i=1}^4 b_ia_i^n$ for the number of partitions of $n$.

Yuval Filmus
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  • I'm sorry, but what is the generating series? – math_sucker Sep 10 '14 at 19:35
  • Sorry, should have been generating function. – Yuval Filmus Sep 10 '14 at 19:37
  • No, that's not the problem. The problem is I don't know what generating function is nor do I know how to solve it. I'm sorry, but, care to explain? Thank you so much. – math_sucker Sep 10 '14 at 19:39
  • Generating functions are the tool you need to solve your exercise. There is ample information available on the web, for example on Wikipedia. Try there first and come back here only if you still cannot understand. – Yuval Filmus Sep 10 '14 at 19:41
  • Okay, I just did what you told me. I've read http://en.wikipedia.org/wiki/Generating_function but I still don't know how to find particular coefficient. Any references to read? Thank you. – math_sucker Sep 10 '14 at 19:44
  • @math_sucker Explained how to obtain a formula in your particular case. – Yuval Filmus Sep 10 '14 at 20:06
  • I think I kinda figured it out. But I have to make sure of this. Anyway, thank you so much for helping me. I'll come back in a couple of dozen minute. – math_sucker Sep 10 '14 at 20:13
  • So, after some trials, turns out finding the particular coefficient is a lot similar to finding a term in fibonacci sequence. Thank you for your clues! – math_sucker Sep 10 '14 at 20:30