How many different piles?

Question

From an idea I saw on twitter. How many different ways can you use 15 objects, separated into 4 four piles with none of the piles having the same number?

Answer 1

The answer to your example is 432, from brute force enumeration. The question gets a little complicated because the numbers in each pile are not independent: The numbers depend on those that came before, both for overlap and the sum to 15.