Using nested multiple summations with a Random function.

Question

I am using a multiple variable summation formula, but I cannot seem to render it in mathematical text.

The principle is as follows: You have five hundred thousand units, they can be stored in any sized allotments, so in theory you could have five hundred thousand allotments (easy) to fifty allotments of variable potential sizes. The number of allotments is a Rand function, and the size of each allotment is a Rand function where they add up to the total units precisely.

Answer 1

https://brilliant.org/wiki/partition-of-an-integer/

This is close, but not quite.

In the example cited they show

• 5
• 4 | 1
• 3 | 2
• 3 | 1 | 1
• 2 | 2 | 1
• 2 | 1 | 1 | 1
• 1 | 1 | 1 | 1 | 1

Where I am of the need to process it like this:

• 5
• 4 | 1
• 1 | 4
• 3 | 2
• 2 | 3
• 3 | 1 | 1
• 1 | 3 | 1
• 1 | 1 | 3
• 2 | 2 | 1
• 2 | 1 | 2
• 1 | 2 | 2
• 2 | 1 | 1 | 1
• 1 | 2 | 1 | 1
• 1 | 1 | 2 | 1
• 1 | 1 | 1 | 2
• 1 | 1 | 1 | 1 | 1

So where the Partitions of an Integer show a value of 7, my 'variation' would show 16.

It appears this is a new direction in Partitions of an Integer, therefore the answer is not in existing math.