Probably the last "7" in the problem is meant to be an "$ n$" instead. In that case, this problem asks for the number of indecomposable permutations. These are counted by A003319 in the OEIS. One way to count these is as follows: Every permutation can be broken up uniquely as a maximal indecomposable parts followed by some other permutation. For example, for the permutation $ 1423756$ we have the decomposition $ 1 - 423756$, while for the permutation $ 2431756$ we have the decomposition $ 2431 - 756$ and for $ 7562431$ we have the decomposition $ 7562431 - \emptyset$. Thus, if $ a_i$ is the number of indecomposable permutations of length $ i$, we have that $ n! = \sum_{i = 1}^n a_i \cdot (n - i)!$. This allows us to quickly compute \begin{align*} a_1 & = 1! = 1 \\ a_2 & = 2! - a_1 \cdot 1! = 1 \\ a_3 & = 3! - a_1 \cdot 2! - a_2 \cdot 1! = 3 \\ a_4 & = 4! - a_1 \cdot 3! - a_2 \cdot 2! - a_3 \cdot 1! = 13 \end{align*} and so-on. One can easily extend this analysis to get the generating-functional results mentioned in the links above. (Note that the generating functions in question are ordinary generating functions, and so exist in a purely formal sense.)