The Euler numbers $f(n,k)$ are given in Generatingfunctionology as the number of permutations of $[n]$ with exactly $k$ increasing runs (exercise 1.18.c). Its recurrence relation is $$f(n,k) = kf(n-1, k) + (n-k+1)f(n-1,k-1)$$ but I don't understand why the second term is multiplied by $(n-k+1)$.
My reasoning is as follows: there are $n$ positions you could stick the $n$ in a permutation of $[n-1]$. Of these, when adding to $k$ positions (ends of the increasing sequences) we will still have $k$ increasing runs $n$, while in the remaining $n-k$ positions we will get one more increasing sequence. Thus $f(n-1,k-1)$ should be multiplied by $n-k$.
What is wrong with this reasoning?