bijection between lattice path and permutations

Question

I am studying Viennot's combinatorial model for the Laguerre polynomials this semester under the guidance of my math professor. If I understand correctly, a bijection exists between the number of Laguerre lattice paths of length $n$, $(n+1)!$, and the symmetric group of permutations on $n+1$. My question is: If there are, say, $24$ paths, then does that mean there are also $24$ permutations? Does a bijection imply the same number of ways for both? Any clarification would be helpful.

Answer 1

Yes. Your bijection gives you a map from the Laguerre lattice paths to the elements of the symmetric group. This map is both injective and surjective, so each path is associated with a distinct element of $S_{n+1}$, and there is some lattice path that is sent to every element of $S_{n+1}$.

Colloquially, this shows that you always have the same number of each because you have a function which "pairs up" the lattice paths with permutations and leaves nothing left over. So, there is always the same number of both. By knowing how many of either exist, you see that this is the number of the other.