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Concept
Dobiński's formula
Summary
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.
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.
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.
Dobiński's formula can be seen as a particular case, for , of the more general relation:
and for in this formula for Touchard polynomials
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 {ƒ −1(b) : b ∈ B} 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 pi that is the kernel. The number of one-to-one functions from pi into B is (x)|pi|, where |pi| is the number of parts in the partition pi.
Official source
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