How many loop-free, weakly-connected digraphs of n vertices are there whose vertices all have indegree 1?
Here are two examples of such digraphs with $n = 5$:
- $v_1 \to \{v_2, v_3, v_4, v_5\}; \; v_5 \to v_1$
- $v_1 \to v_2; \; v_2 \to v_3; \; v_3 \to v_4; \; v_4 \to v_5; \; v_5 \to v_1$
Is there a theorem or formula that describes the number of these digraphs that exist for $n$ vertices, up to isomorphism?
Update: A commenter asked for some background.
I'm writing a puzzle for a game which the player must solve. The player activates a series of beam emitter-receivers (BER) in various positions. Each BER can receive only one beam but can emit as many as it wants to the other BERs.
The puzzle is solved when every BER is receiving energy from some other BER. I'm curious about the number of combinations that are possible with an n-instance configuration of BERs, so I asked this question.