3

In the jungle of posts on math.stackexachange related to this subject I have been searching quite a while now. I read some very useful posts however I can't solve my own problem. My version of the number of integer solutions sounds like this:

Question: Let $x + y + z \leq 13$ with the following restrictions: $x \leq y \leq x+2$ and $x \leq z$. Find the number of possibilities using a generating function.

Related to this problem I found this post. I tried so for the equation $x + y + z =13$:

Say $z=x+z'$, $x+2=y+x'$ and $y=x+y'$. However replacing those expressions into the equation, I got stuck with replacing $x'$ and $y'$ for $x$ and $y$. The only thing I see is that: $x' + y'=2$. How can I use this technique properly?

And what about the inequality? How to deal with $x + y + z \leq 13$? Thanx in advance!

Community
  • 1
iJup
  • 1,999
  • Are $x,y,z$ non-negative integers? – user26486 Feb 22 '15 at 08:47
  • 3
    We deal with the inequality part only, in order to mention useful trick. The number of solutions in non-negative integers of $x_1+\cdots +x_n\le k$ is equal to the number of solutions in non-negative integers of $x_1+\cdots +x_n+x_{n+1}=k$. – André Nicolas Feb 22 '15 at 08:48
  • Yes of course! I forgot to mention this, sorry. – iJup Feb 22 '15 at 08:50

1 Answers1

4

As you say, the inequalities are essentially just saying that $y=x+y', x+2=y+x', z=x+z'$ for some non-negative integers $x',y',z'\in\mathbb Z_{\ge 0}$. Then:

$$3x+y'+z'\le 13\iff 3x+y'+z'+A=13$$

for some non-negative integer $A$.

But we still haven't used $x+2=y+x'\iff x'+y'=2$ (we used $y=x+y'$), which is just saying that $y'\in[0;2]$.

Thus the generating function we need is:

$$(1+x^3+x^6+x^9+x^{12})(1+x+x^2)(\sum_{i=0}^{13} x^i)(\sum_{i=0}^{13} x^i)$$

The coefficient of $x^{13}$ will be our answer. It is 105.
For a way of finding the coefficient of generating functions see this answer.

Since $(1+x^3+x^6+x^9+x^{12})(1+x+x^2)=(1+x+x^2+\cdots+x^{14})$, we can simply write our generating function as:

$$\left[\left(\sum_{i=0}^\infty x^i\right)^3\right]_{x^{13}}=\left[\frac{1}{(1-x)^3}\right]_{x^{13}}=\binom{13+3-1}{13}=\binom{15}{13}=\binom{15}{2}=\frac{15\cdot 14}{2}=15\cdot 7=105$$

Community
  • 1
user26486
  • 11,331