Enumerate all lists of $m$ non-negative integers that add to $n$

Asked Sep 24 '18 at 15:31

Active Sep 24 '18 at 16:01

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Let $x_1,\dots,x_m$ be non-negative integers such that $\sum_{i=1}^m x_i=n$, where $m,n$ are given. How can I enumerate all such lists of $m$ integers that add to $n$?

Note that IntegerPartitions[n,{m}] counts two such lists as one if they are a permutation of each other, but I would count them as distinct.

Thus the list I want must have $\binom{m+n-1}{m-1}$ elements.

edited Sep 24 '18 at 16:01

asked Sep 24 '18 at 15:31

a06e

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Apply[Sequence]@*Permutations /@ IntegerPartitions[n, {k}] – AccidentalFourierTransform Sep 24 '18 at 15:37
Try FrobeniusSolve[ConstantArray[1, m], n]. (I am sure this is a dupe.) – J. M.'s missing motivation Sep 24 '18 at 15:38
@AccidentalFourierTransform Thanks, that works. I didn't know that Permutations considered repeated elements as identical. Feel free to post an answer. – a06e Sep 24 '18 at 15:42
@J.M.issomewhatokay. Thanks, also works. But AccidentalFourierTransform's answer seems faster. – a06e Sep 24 '18 at 15:42
Did you want non-negative integers or positive integers? The two suggestions in the comments are different. – Carl Woll Sep 24 '18 at 15:47
Check the Q&A's linked from this and tell me if you find one that matches your need: https://mathematica.stackexchange.com/q/101046/121 – Mr.Wizard Sep 24 '18 at 15:51
@CarlWoll You are right, I didn't notice. I wanted non-negative integers, so actually J.M. solution is the correct one. Although the other solution can be modified to Apply[Sequence]@*Permutations /@ (IntegerPartitions[n + k, {k}] - 1) – a06e Sep 24 '18 at 15:52
Based on your comment immediately above I have marked this as I believe is correct. – Mr.Wizard Sep 24 '18 at 15:54
@Mr.Wizard Yes, I missed that. Thanks. – a06e Sep 24 '18 at 15:55

Enumerate all lists of $m$ non-negative integers that add to $n$

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