The number of ways a set of 
elements can be partitioned
into nonempty subsets is called a Bell number and is denoted
(not to be confused with the Bernoulli
number, which is also commonly denoted 
).
For example, there are five ways the numbers 
can be partitioned: 
, 
, 
, 
, and 
, so 
.
, and the first few Bell numbers
for 
, 2, ... are 1, 2, 5, 15, 52, 203,
877, 4140, 21147, 115975, ... (OEIS A000110).
The numbers of digits in 
for 
, 1, ... are given by 1, 6, 116, 1928,
27665, ... (OEIS A113015).
Bell numbers are implemented in the Wolfram Language as BellB[n].
Though Bell numbers have traditionally been attributed to E. T. Bell as a result of the general theory he developed in his 1934 paper (Bell 1934), the first systematic study of Bell numbers was made by Ramanujan in chapter 3 of his second notebook approximately 25-30 years prior to Bell's work (B. C. Berndt, pers. comm., Jan. 4 and 13, 2010).
The first few prime Bell numbers occur at indices 
, 3, 7, 13, 42, 55, 2841, ... (OEIS A051130),
with no others less than 
(Weisstein, Apr. 23, 2006). These correspond to the numbers 2, 5, 877, 27644437,
... (OEIS A051131). 
was proved prime by I. Larrosa Canestro in 2004
after 17 months of computation using the elliptic
curve primality proving program PRIMO.
Bell numbers are closely related to Catalan numbers. The diagram above shows the constructions giving 
and 
, with line segments representing elements in the same
subset and dots representing subsets containing a single
element (Dickau). The integers 
can be defined by the sum
 |
(1)
|
where 
is a Stirling number of the second
kind, i.e., as the Stirling transform of
the sequence 1, 1, 1, ....
The Bell numbers are given in terms of generalized hypergeometric functions by
 |
(2)
|
(K. A. Penson, pers. comm., Jan. 14, 2007).
The Bell numbers can also be generated using the sum and recurrence relation
 |
(3)
|
where 
is a binomial coefficient, using the formula
of Comtet (1974)
![B_n=[e^(-1)sum_(m=1)^(2n)(m^n)/(m!)]](/images/equations/BellNumber/NumberedEquation4.svg) |
(4)
|
for 
, where ![[x]](/images/equations/BellNumber/Inline24.svg)
denotes the ceiling function.
Dobiński's formula gives the 
th Bell number
 |
(5)
|
A variation of Dobiński's formula gives
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
is a subfactorial (Pitman 1997).
A double sum is given by
 |
(8)
|
The Bell numbers are given by the generating function
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  | ![((-1)^(1/x)[xGamma(1-x^(-1))+Gamma(-x^(-1),-1)])/(ex)](/images/equations/BellNumber/Inline44.svg) |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
and the exponential generating function
 |
(15)
|
An amazing integral representation for 
was given by Cesàro (1885),
 |  | ![(2n!)/(pie)I[int_0^pie^(e^(e^(ntheta)))sin(ntheta)dtheta]](/images/equations/BellNumber/Inline54.svg) |
(16)
|
 |  | ![(2n!)/(pie)int_0^pie^(e^(costheta)cos(sint))sin[e^(costheta)sin(sintheta)]sin(ntheta)dtheta](/images/equations/BellNumber/Inline57.svg) |
(17)
|
(Becker and Browne 1941, Callan 2005), where ![I[z]](/images/equations/BellNumber/Inline58.svg)
denotes the imaginary part
of 
.
The Bell number 
is also equal to 
,
where 
is a Bell polynomial.
de Bruijn (1981) gave the asymptotic formula
![(lnB_n)/n=lnn-lnlnn-1+(lnlnn)/(lnn)+1/(lnn)+1/2((lnlnn)/(lnn))^2+O[(lnlnn)/((lnn)^2)].](/images/equations/BellNumber/NumberedEquation8.svg) |
(18)
|
Lovász (1993) showed that this formula gives the asymptotic limit
![B_n∼n^(-1/2)[lambda(n)]^(n+1/2)e^(lambda(n)-n-1),](/images/equations/BellNumber/NumberedEquation9.svg) |
(19)
|
where 
is given by
 |
(20)
|
with 
the Lambert
W-function (Graham et al. 1994, p. 493). Odlyzko (1995) gave
 |
(21)
|
Touchard's congruence states
 |
(22)
|
when 
is prime.
This gives as a special case for 
the congruence
 |
(23)
|
for 
prime. It has been conjectured that
 |
(24)
|
gives the minimum period of 
(mod 
). The sequence of Bell numbers 
is periodic (Levine and Dalton 1962, Lunnon et
al. 1979) with periods for moduli 
, 2, ... given by 1, 3, 13, 12, 781, 39, 137257, 24, 39,
2343, 28531167061, 156, ... (OEIS A054767).
The Bell numbers also have the curious property that
 |  |  |
(25)
|
 |  |  |
(26)
|
(Lenard 1992), where the product is simply a superfactorial and 
is a Barnes
G-function, the first few of which for 
, 1, 2, ... are 1, 1, 2, 12, 288, 34560, 24883200, ... (OEIS
A000178).
, of First Order Bell Exponential Integrals." Math.
Comput. 16, 416-423, 1962.Lovász, L.