In combinatorial mathematics, Dobiński's formula[1] states that the
, the number of partitions of a set of size
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
th 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
th 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
(a condition that of course implies
Dobiński's formula can be seen as a particular case, for
in this formula for Touchard polynomials
One proof[2] relies on a formula for the generating function for Bell numbers,
The power series for the exponential gives
Another style of proof was given by Rota.
are nonnegative integers then the number of one-to-one functions that map a size-
set is the falling factorial
Rota calls this partition the "kernel" of the function
factors into The first of these two factors is completely determined by the partition
The number of one-to-one functions from
is the number of parts in the partition
Thus the total number of functions from a size-
running through the set of all partitions of
On the other hand, the number of functions from
Rota continues the proof using linear algebra, but it is enlightening to introduce a Poisson-distributed random variable
th moment of this random variable is
and this is just the number of partitions of the set
th factorial moment of the random variable
is a Poisson-distributed random variable with mean 1, recall that this random variable assumes each value integer value