In combinatorial mathematics, Dobiński's formula states that the n-th Bell number Bn (i.e., the number of partitions of a set of size n) equals where denotes Euler's number. The formula is named after G. Dobiński, who published it in 1877. Probabilistic content In the setting of probability theory, Dobiński's formula represents the nth moment of the Poisson distribution with mean 1. Sometimes Dobiński's formula is stated as saying that the number of partitions of a set of size n equals the nth moment of that distribution. Reduced formula The computation of the sum of Dobiński's series can be reduced to a finite sum of terms, taking into account the information that is an integer. Precisely one has, for any integer provided (a condition that of course implies , but that is satisfied by some of size ). Indeed, since , one has Therefore for all so that the tail is dominated by the series , which implies , whence the reduced formula. Generalization Dobiński's formula can be seen as a particular case, for , of the more general relation: Proof One proof relies on a formula for the generating function for Bell numbers, The power series for the exponential gives so The coefficient of in this power series must be , so Another style of proof was given by Rota. Recall that if x and n are nonnegative integers then the number of one-to-one functions that map a size-n set into a size-x set is the falling factorial Let ƒ be any function from a size-n set A into a size-x set B. For any b ∈ B, let ƒ −1(b) = {a ∈ A : ƒ(a) = b}. Then } is a partition of A. Rota calls this partition the "kernel" of the function ƒ. Any function from A into B factors into one function that maps a member of A to the element of the kernel to which it belongs, and another function, which is necessarily one-to-one, that maps the kernel into B. The first of these two factors is completely determined by the partition that is the kernel. The number of one-to-one functions from into B is (x)||, where || is the number of parts in the partition . Thus the total number of functions from a size-n set A into a size-x set B is the index running through the set of all partitions of A.