Let A represent the number of labeled vertices on the left side of a bipartite graph, and let B represent the number of labeled vertices on the right. How many ways are there to connect every vertex in A to some number of vertices in B such that no vertex in B is left behind?
You are only given A and B, the number of labeled vertices in the left and right set. I am asking how many ways there are to place edges such that no vertex is left untouched.
Edit: For some reason I can't reply to the messages below, so I have to do it here. I don't think the current answers are right. By exhaustion I count 7 such cases for A=B=2, whereas the answers return 9.
Edit again: Look at the upper left and lower left graphs of the pictures posted below in Fabrício Caluza Machad's answer. Some vertices don't have any edges connecting to them. This is what I mean by "no vertex untouched."