On the other hand, the number of functions from A into B is clearly xn. Therefore, we have Rota continues the proof using linear algebra, but it is enlightening to introduce a Poisson-distributed random variable X with mean 1. The equation above implies that the nth moment of this random variable is where E stands for expected value. But we shall show that all the quantities E((X)k) equal 1. It follows that and this is just the number of partitions of the set A. The quantity E((X)k) is called the kth factorial moment of the random variable X. To show that this equals 1 for all k when X is a Poisson-distributed random variable with mean 1, recall that this random variable assumes each value integer value with probability . Thus Notes and references Combinatorics Poisson point processes Articles containing proofs