Is there a way to calculate the number of constrained integer partitions containing particular summands? By constrained, I mean, the permitted summands must be below a certain limit, such as 5. Take the number 1,000,000, for example.
I'd like to know how many sums forming this number, involve…
- A single summand 1 (e.g. $1 + 2 + 3 \cdot (333332)$)
- Two summands 1s (e.g. $1 + 1 + 2 \cdot (499999)$)
- …
- Two summands 5s (e.g. $5 + 5 + 3 \cdot (10) + 4 \cdot (249990)$)
- …
Ideally, I'd love to create a grid in which each row is the summand of interest, and each column is the count of 1,000,000's partition summands matching the row number. This grid would look jagged, given that the maximum number of ways for summands to be contained in a partition is decreasing with the size of the summand:
- Row 1 will have 1,000,000 rows.
- Row 2 will have 500,000 rows.
- Row 5 will have 200,000 rows, and so on.
Is anyone aware of a standard mathematical process for doing the above?