Counting certain elements of a generalized symmetric group

Question

Let $G$ be the wreath product of the cyclic group of order $n$ with the symmetric group $S_r$. The group $G$ acts on the set $X=\{1,\ldots, n\}\times\{1,\ldots, r\}$ in a natural way. How many elements of $G$ move every element of $X$?

Answer 1

The number $f(n,r)$ you are looking for should be $$f(n,r)=\sum_{i=0}^r\;(n-1)^i\cdot n^{r-i}\cdot D(r, i)$$ where $D(r,i)$ is the number of partial derangements, i.e. the number of permutations $\in S_r$ fixing exactly $i$ elements.