I want to write a formula for $n!$.
$n!$ is the number of permutation functions on the set $\{1,\ldots,n\}$.
Let's define a "true k-permutation" on $\{1,\ldots,n\}$ as a permutation that is identity on exactly $n-k$ elements, i.e., if $\phi$ is a $k$-permutation then $\phi(i) = i$ only on $n-k$ elements.
This seems like an elementary object yet I don't know what you call such perms (That is my question).
Notice that the relation $\phi$ is a true $k$-permutation partitions the group of permutations on $\{1,\ldots,n\}$.
So there is a formula for $n!$ involving counting of true $k$-perms. The one I have involves sum partitions of $k$ and disjoint compositions of cycles. But there's probably other ways to count them.
But this has to have been done already, so I would appreciate it if someone could post a link.