Integer Partition Calculator

Unrestricted p(n) is deliberately computed with the part-size cap reset to n itself and distinctness switched off every time, regardless of whatever Max Part Size or Distinct Parts Only you've set for the main Partition Count. It exists to show the baseline count with no constraints at all, so your constraint settings never reach that particular calculation.

Growing n opens up an entire new range of achievable sums and combinations, so the partition count grows combinatorially with it. Max Part Size, by contrast, only ever affects whether partitions using a part right at (or above) that cap are counted or excluded, a much narrower slice of the total than n's effect on the whole combinatorial space.

With it off, the calculator allows any part size to repeat as many times as needed to reach the sum — so 4 could be written as 2 + 2. With it checked, every part in a counted partition must be a different value from every other part in that same partition, which rules out sums like 2 + 2 and generally produces a smaller partition count for the same n.

Setting Max Part Size equal to n places no real restriction at all, since no single part in a partition of n can ever exceed n itself anyway — every partition remains eligible. That's exactly the setting the calculator falls back to automatically whenever Max Part Size is left at its default, which is why the default Partition Count and Unrestricted p(n) typically agree.