Let's say I'm faced with the problem of counting the number of elements in a set A. This set is extremely difficult to count, but maybe I could set a bijection with another known and well studied set (such as the ones that can be computed using Catalan numbers or Euler numbers).
Of course at first I could only do this by trial and error since it's very difficult to see a bijection immediately. So, which sets are the most common ones I could try?
For example, consider the problem of finding the number of lists that have N elements where every element is a number between 1 and K inclusive, and for every element i greater than 1, i-1 appears before the last occurrence of i in the list. The cardinality of this set can be found using Euler numbers.
So the question is, what are the most common sets that I should try to biject to first?