Write Partition of $n$ as an $n$-tuple

Question

I recently asked the question (Computationally Employ Sum over Ordered Partitions?) which I think I can work through. But there's a subtelty I overlooked before that problem is even well-defined, which is maybe deserving of its own question.

I'm interested in considering ordered partitions of an integer $n$. However, using a built in function like IntegerPartitions the output will be in the form, for example:

{{3,0},{2,1},{1,2},{1,1,1}}

What I would like to do is have Mathematica write these ordered partitions as $n$-tuples. What I mean by this is best illustrated with an example. Consider $n=3$ objects, for a small $n$ example. There are two slots between the objects where we can place barriers.

$$\{\, \cdot \, | \, \cdot \, | \, \cdot \}$$

I would like the above ordered partition to be given in Mathematica by {1,1,1}.

$$\{\, \cdot \, \, \cdot \, | \, \cdot \}$$

The above partition, I want to write as {2,0,1} because there is no barrier in the first slot.

$$\{\, \cdot \, | \, \cdot \, \, \cdot \}$$

The above I want to write as {1,2,0} and of course...

$$\{\, \cdot \, \, \cdot \, \, \cdot \}$$

this I want to write as {3,0,0}. Is there a way I can manipulate Mathematica's built-in function IntegerPartitions to do this for any $n$?

Answer 1

orderedPartitions[n_] := Flatten[#, 1] &@(Permutations /@ IntegerPartitions[n]);
paddedOrderedParitions[n_] := Flatten /@ (orderedPartitions[n] 
/. {x_Integer /; x != 1 :> PadRight[{x}, x]});

n=3,

{{3, 0, 0}, {2, 0, 1}, {1, 2, 0}, {1, 1, 1}}

n=5,

{{5, 0, 0, 0, 0}, {4, 0, 0, 0, 1}, {1, 4, 0, 0, 0}, {3, 0, 0, 2, 0}, {2, 0, 3, 0, 0}, {3, 0, 0, 1, 1}, {1, 3, 0, 0, 1}, {1, 1, 3, 0, 0}, {2, 0, 2, 0, 1}, {2, 0, 1, 2, 0}, {1, 2, 0, 2, 0}, {2, 0, 1, 1, 1}, {1, 2, 0, 1, 1}, {1, 1, 2, 0, 1}, {1, 1, 1, 2, 0}, {1, 1, 1, 1, 1}}