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How many nonnegative integer solutions of the following diophantine equations? And how to find them? \begin{equation} \left\{ \begin{aligned} x_1+x_2+\cdots+x_n=r\\ x_1+2x_2+\cdots+nx_n=n\\ \end{aligned} \right. \end{equation} where $1\leq r\leq n;r,n\in Z^+.$

I will appreciate it for any advices and suggestions! Thank you very much!

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  • $\begingroup$ @Matt Solutions are only required to be non-negative, so I don't see even why the LHS of the first equation has to be at least $n$. $\endgroup$ – provocateur Oct 21 '19 at 12:31
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The solutions correspond to partitions of $n$ into $r$ parts, and $x_i$ is the number of times that part $i$ is used. A recurrence is given here, and you can use a similar recursion to enumerate them.

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  • $\begingroup$ @Pratt Thanks a lot. Actually, it is equivalent to the following equations: \begin{equation} \left\{ \begin{aligned} x_1+x_2+\cdots+x_{n-r+1}=r\\ x_1+2x_2+\cdots+(n-r+1)x_{n-r+1}=n\\ \end{aligned} \right. \end{equation} where $1\leq r\leq n;r,n\in Z^+.$ $\endgroup$ – Jacob.Z.Lee Oct 23 '19 at 1:41
  • $\begingroup$ There are some relations between patial Bell Polynomial and this problems. The number of solutions is the number of monomials in Bell polynomial $B_{n,r}(x_1,x_2,\cdots, x_{n-r+1})$. $\endgroup$ – Jacob.Z.Lee Oct 23 '19 at 1:49
  • $\begingroup$ @Pratt. The recurrence just gives the number of solutions. My question is how to find these solutions. $\endgroup$ – Jacob.Z.Lee Oct 23 '19 at 1:52
  • $\begingroup$ Condition on $x_n$, the number of times that part $n$ appears. $\endgroup$ – RobPratt Oct 23 '19 at 6:28

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