How many directed graphs of size n are there where each vertex is the tail of exactly one edge? In a research problem in an unrelated area, me and a student found it necessary to count the number of directed graphs with every vertex having one outward-pointing edge, with no restrictions on the number of edges pointing inward.
Not being a graph theorist, I was wondering if this is a well-known class of directed graphs. 
Is the number of such graphs for fixed n known?
If it is unknown or an open problem, we'll work on it together, but it seems likely to be known.
 A: A beginning:
Let $f:\>V\to V$ be the map defined by $f(v):=$ endpoint of the edge emanating from $v\in V$. 
For an intuitive understanding of what's going on here you should consider the iterates $f^{\circ k}$, $\>k\geq0$. Since $V$ is finite each orbit $$O(v):=\{f^{\circ k}(v)\>|\>k\geq0\}$$
must end in a cycle. Call two vertices equivalent if their orbits end in the same cycle. The equivalence classes, together with the connecting edges,  are the components of your graph $\Gamma$. 
Such a component has the following shape: There is a final cycle $\gamma=\{v_1,v_2,\ldots, v_r\}$ of length $r\geq1$ if loops are allowed in $\Gamma$, and  $r\geq2$ otherwise. Each $v_i\in\gamma$ is the root of a rooted tree collecting the vertices whose orbits enter $\gamma$ at $v_i$.
I hope that such things can be counted up to isomorphism using Polya counting theory. (The counting of rooted trees is a standard example in this theory. But here the situation is more complicated, since we have necklaces of rooted trees.)
