Scenario 1) There are 21 identical books. How many ways to place the books into 2 indistinguishable piles? (This means that putting 16 books in the first pile and 5 books in the second is the same as 5 books in the first pile and 16 books in the second)
Scenario 2) There are 21 books, all different and 2 distinct shelves. How many ways are there to place the books on the shelves?
For scenario one, the number of ways to place the books in indistinguishable piles is the number partitions of the number 21 (the number of ways you can add numbers to get 21). This is 792, which can be found from - https://www.dcode.fr/partitions-generator. There's not a way you can analytically calculate it, you can try approximating it with things like the Hardy-Ramanujan formula, but there is no exact function that I know of. If there were a specified number of piles, this would be a lot easier.
For scenario 2, each book can either be placed on shelf 1 or 2, so the total number of combinations is just $2^{21}$ or 2,097,512 ways.