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I would like to write a function that would, given a positive integer $n$, compute the following:

$S_n = \sum_{(x,y,z)} a_{(x,y,z)} f(x,y,z)$, where the sum runs over all tuples $(x,y,z)$ such that $x+y+z = n$, and $x,y,z$ take values in {$0,1,...,n$}. The ultimate goal is to pass this function trough Solve to determine the coefficients $a_{(x,y,z)}$ with some other conditions from the problem.

My main problem is generating the appropriate tuples in a way that I could then pass through Sum.

Any help is appreciated.

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    $\begingroup$ Have you already seen FrobeniusSolve[]? $\endgroup$ – J. M.'s torpor May 26 '20 at 17:41
  • $\begingroup$ I have not, I'll look it up $\endgroup$ – johnny May 26 '20 at 17:46
  • $\begingroup$ Yep, it looks like it works, thanks! $\endgroup$ – johnny May 26 '20 at 17:55
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    $\begingroup$ @johnny If you have a solution in hand, please write up an answer to your question, so it does not remain dangling for future users. $\endgroup$ – MarcoB May 26 '20 at 18:11
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    $\begingroup$ As @J.M. says, FrobeniusSolve[ConstantArray[1, n], n] would do; but if you can make any use of permutational equivalence, then IntegerPartitions[n, {n}, Range[0, n]] may also help as it generates a vastly smaller list (which needs to be permuted, however). $\endgroup$ – Roman May 26 '20 at 20:03
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As suggested by @MarcoB, here is my solution, using FrobeniusSolve:

Frobenius[n_] := FrobeniusSolve[{1, 1, 1}, n];
Sol[n_] :=
Sum[a[Part[Frobenius[n], i] /. List -> Sequence]
*OrderN[Part[Frobenius[n], i] /. List -> Sequence],
{i, 1, Length[Frobenius[n]]}]

In the code, corresponds to the $f(x,y,z)$ in the original question.

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  • $\begingroup$ Thank you for writing this up! (+1) $\endgroup$ – MarcoB May 28 '20 at 5:25

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