Functions with certain properties

I've been trying to figure out the general answer to this question. It seems like I should be able to sort it out, but I've just got a mental block.

For a set of values N, how many functions f exist, such that the following are true? (Forgive me if I make mistakes with the notation...I'm not a mathematician.)

f ( f (a, b), c) = f ( f (a, c), b)

∃ g : g (a, f (a, b)) = b

where a, b, c ← N

I've been able to determine the answer through brute-force exploration of the function space for the following sizes of N:

For |N| = 2, there are 2 functions with these properties.

For |N| = 3, there are 6 functions with these properties.

For |N| = 4, there are 96 functions with these properties.

It seems like it ought to be easy to say, for a given size |N|, how many of these functions there are. But I can't see it yet.

Note 12/3/2013: I figured it out.