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Generating Functions

This week, we have a guest post written by Alexander B. on generating functions, which are useful tools for solving advanced counting problems.

How would you use generating functions to solve the following?

>

  1. How many non-negative integer solutions are there to \( a + b + 2c = n \) for any positive integer \(n\)?

  2. Share a question which can be approached using Generating Functions.


If you are interested in contributing a post, please contact me at Calvin@Brilliant.org.

Note by Calvin Lin
4 years ago

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After you finish number 1, try this:

How many non-negative integer solutions are there to \(a+b+2c=n\) for any positive integer \(n\) for distinct \(a, b, c\)?

Logan Dymond - 4 years ago

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If \(d_n\) is the number of solutions with \(a=b\), then \[ d_n \; = \; \left\{ \begin{array}{lcl} \tfrac12n+1 & \quad & n \mbox{ even} \\ 0 & \quad & n \mbox{ odd} \end{array}\right. \] If \(e_n\) is the number of solutions with \(a=c\) or \(b=c\) then \[ e_n \; = \; 1 + \big\lfloor \tfrac13n\big\rfloor \] If \(f_n\) is the number of solutions with \(a=b=c\) then \[ f_n \; = \; \left\{ \begin{array}{lcl} 1 & \quad & 4\,|\,n \\ 0 & \quad & 4 \not| n \end{array} \right. \] and the number of solutions with \(a,b,c\) distinct is \[ c_n - d_n - 2e_n + 2f_n \] where \(c_n = \big\lfloor \tfrac14(n+2)^2\big\rfloor\) is as in my other post.

Mark Hennings - 4 years ago

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The number of solutions, \(c_n\), is the coefficient of \(x^n\) in the expansion of \[ \frac{1}{(1-x)^2(1-x^2)} \; = \; \frac{1}{(1-x)^3(1+x)} \; = \; \left(\sum_{j=0}^\infty {j+2 \choose 2}x^j\right)\left(\sum_{k=0}^\infty (-1)^kx^k\right) \] and so \[ \begin{array}{rcl} c_n & = & \sum_{m=0}^n (-1)^{n-m}{m+2 \choose 2} \; = \; \tfrac12(-1)^n\sum_{m=0}^n(-1)^m(m+1)(m+2) \\ & = & \tfrac14(-1)^n\sum_{m=0}^n (-1)^m\big[(m+1)^2+(m+2)^2-1\big] \\ & = & \tfrac14(-1)^n\left[\sum_{m=0}^n (-1)^m(m+1)^2 - \sum_{m=1}^{n+1}(-1)^m(m+1)^2 - \sum_{m=0}^n(-1)^m\right] \\ & = &\tfrac14(-1)^n\left[1 + (-1)^n(n+2)^2 - \sum_{m=0}^n(-1)^m\right] \; = \; \tfrac14\big[(n+2)^2 - \varepsilon_n\big] \end{array} \] where \[ \varepsilon_n \; = \; \left\{ \begin{array}{lcl} 0 & \qquad & n \mbox{ even} \\ 1 & \qquad & n \mbox{ odd} \end{array} \right. \] and hence \[ c_n \; = \; \left \lfloor \tfrac14(n+2)^2 \right\rfloor \]

Mark Hennings - 4 years ago

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Awesome..Till now, I was completely unknown to this concept,(I had seen it in problems but never understood )but now I am very interested in it...I am very happy now..thnx Alexander and Calvin.

Kishan K - 4 years ago

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Can you use generating functions when you are picking 2 or more distinct elements from one set?

Michael Tong - 4 years ago

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Consider \( f(x)^2 - f(x^2) \).

What would it look like for picking 3 distinct objects?

Calvin Lin Staff - 4 years ago

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If \(f(x) = x^4 + x^5 + x^6 + x^7\) then it would be \((f(x))^3 - x^4 f(x^2) - x^5 f(x^2) - x^6f(x^2) - x^7f(x^2)\)?

Mike Kong - 4 years ago

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@Mike Kong Write it out and check :)

Your formula looks like \( f(x)^3 - f(x) f(x^2) \).
If it works, why does it work?
If it doesn't work, why doesn't it work?

Calvin Lin Staff - 4 years ago

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Ah, brilliant. Though as you start picking 3, 4, 5, etc. elements this doesn't become very practical anymore.

To your edit-- It would be \(f(x)^3 - f(x^3) - \) some other things to get rid of the products in which only a pair are the same. I'll think about it.

Michael Tong - 4 years ago

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@Michael Tong There is a 'nice' generalized formula. Think about how I created the function, and why it works.

I don't think your example works.

Hint: Just because you learnt an advanced technique, doesn't mean that you should forget your basics.

Calvin Lin Staff - 4 years ago

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@Calvin Lin I know mine doesn't work, my sentence comes out weird in latex but what I meant was "this minus some other things." Your first one works because you take the product and then subtract the case when two of the same element is being multiplied by each other, but when you scale up to three you have to worry about all three of them being the same as well as them being not pairwise distinct (e.g. \(x^4 \times x^4 \times x^7\)). However, while a term being multiplied three times occurs only once, the situation where two terms plus another distinct term occurs thrice. This makes me come to the formula of \(f(x)^3 - 3f(x)f(x^2) + 2f(x^3)\). It's multiplying by three, then taking out three times the pairwise non-distinct as well as the triple non-distinct, and then adding two of the triple non-distincts back in since that only occurs once in the original expansion of \(f(x)^3\). Though I'm a bit doubtful that this one works either..

Michael Tong - 4 years ago

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@Michael Tong To elaborate on my hint, use the Principle of Inclusion and Exclusion.

This tells you why the formula looks the way it does (and also why it gets ugly quickly). Interestingly, it only involves terms of the form \( f(x^n) \), so if you have a simple description of \( f(x) \), then you might be able to get a simple description for the generating function of distinct elements.

Calvin Lin Staff - 4 years ago

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