Suppose there is a group of $n$ people giving gifts to one another. Everybody brings a gift but we want the gifts to be "well-distributed" in the group. By this I mean the following:

In how many ways can the gifts be given such that - nobody receives their own gift, and in addition - there is no $k$-element subset of the $n$ people where the gifts are exchanged only among these $k$ people, except for $k=n$.

Let me put this in mathematical words:

Let $K_n$ be the complete graph on $n$ vertices, which are labeled with the numbers $1,..,n$. How many Hamiltonian cycles with a specified start vertex $i_1$ are there such that vertex $j$ is not reached at time $j$. I.e., I would like to count ordered sequences $(i_1, ..., i_n)$ with all $i_j\in\{i,..,n\}$ distinct and such that $i_j\neq j$, and the subgraph spanned by the edges $(i_1,1), (i_2,2), ..., (i_n, n)$ is a Hamiltonian path.

One could also express this in terms of group elements $w$ of $S_n$ acting transitively on $\{1,...,n\}$ when taken to powers $w^0$,$w^1,w^2,\dotsc$.