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 2530 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)
(4)

for , where 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)
 
(12)
 
(13)
 
(14)

and the exponential generating function
(15)

An amazing integral representation for was given by Cesàro (1885),
(16)
 
(17)

(Becker and Browne 1941, Callan 2005), where denotes the imaginary part of .
The Bell number is also equal to , where is a Bell polynomial.
de Bruijn (1981) gave the asymptotic formula
(18)

Lovász (1993) showed that this formula gives the asymptotic limit
(19)

where is given by
(20)

with the Lambert Wfunction (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 Gfunction, the first few of which for , 1, 2, ... are 1, 1, 2, 12, 288, 34560, 24883200, ... (OEIS A000178).