From Wikipedia, the free encyclopedia Bell number
Bell number
number,
In combinatorics, the nth Bell number named after Eric { {a}, {b, c} }
Temple Bell, is the number of partitions of a set with n
members, or equivalently, the number of equivalence re- { {b}, {a, c} }
lations on it. Starting with B0 = B1 = 1, the first few Bell
{ {c}, {a, b} }
numbers are:
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, … { {a, b, c} }.
(sequence A000110 in OEIS).
B0 is 1 because there is exactly one partition of the empty
(See also breakdown by number of subsets/equivalence set. Every member of the empty set is a nonempty set
classes.) (that is vacuously true), and their union is the empty set.
Therefore, the empty set is the only partition of itself.
Partitions of a set Note that, as suggested by the set notation above, we
consider neither the order of the partitions nor the order
of elements within each partition. This means the follow-
ing partitionings are all considered identical:
{ {b}, {a, c} }
{ {a, c}, {b} }
{ {b}, {c, a} }
{ {c, a}, {b} }.
Another view of Bell numbers
Bell numbers can also be viewed as the number of dis-
tinct possible ways of putting n distinguishable balls into
one or more indistinguishable boxes. For example, let us
suppose n is 3. We have three balls, which we will label a,
b, and c, and three boxes. If the boxes can not be distin-
guished from each other, there are five ways of putting
the balls in the boxes:
• All three balls go in to one box. Since the boxes are
anonymous, this is only considered one combination.
• goes into one box; b and c go in to another box.
• goes into one box; a and c go in to another box.
• goes into one box; a and b go in to another box.
• Each ball goes in to its own box.
The traditional Japanese symbols for the chapters of the Tale of
Properties of Bell numbers
Genji are based on the 52 ways of partitioning five elements. The Bell numbers satisfy this recursion formula:
In general, Bn is the number of partitions of a set of size
n. A partition of a set S is defined as a set of nonempty,
pairwise disjoint subsets of S whose union is S. For exam-
ple, B3 = 5 because the 3-element set {a, b, c} can be parti- They also satisfy "Dobinski’s formula":
tioned in 5 distinct ways:
{ {a}, {b}, {c} }
1
From Wikipedia, the free encyclopedia Bell number
= the nth moment of a Poisson distribution
with expected value 1.
And they satisfy "Touchard’s congruence": If p is any
prime number then moreover, if then for all n > n0(ε),
or, generalizing
Each Bell number is a sum of Stirling numbers of the sec- where and
ond kind
Triangle scheme for calculating
The Stirling number is the number of ways to par-
Bell numbers
tition a set of cardinality n into exactly k nonempty sub-
sets.
More generally, the Bell numbers satisfy the follow-
ing recurrence[1]:
The nth Bell number is also the sum of the coefficients
in the polynomial that expresses the nth moment of any
probability distribution as a function of the first n cumu-
lants; this way of enumerating partitions is not as coarse
as that given by the Stirling numbers. The triangular array whose right-hand diagonal sequence con-
The exponential generating function of the Bell num- sists of Bell numbers
bers is
The Bell numbers can easily be calculated by creating
the so-called Bell triangle also called Aitken’s array[cita-
triangle,
tion needed] or the Peirce triangle[citation needed]:
1. Start with the number one. Put this on a row by
itself.
Asymptotic limit and bounds 2. Start a new row with the rightmost element from the
Several asymptotic formulae for the Bell numbers are previous row as the leftmost number
known. One such is 3. Determine the numbers not on the left column by
taking the sum of the number to the left and the
number above the number to the left (the number
diagonally up and left of the number we are
calculating)
Here 4. Repeat step three until there is a new row with one
more number than the previous row
5. The number on the left hand side of a given row is
the Bell number for that row.
where W is the Lambert W function. For example, the first row is made by placing one by it-
(Lovász, 1993) self. The next (second) row is made by taking the right-
In (Berend, D. and Tassa, T., 2010), the following most number from the previous row (1), and placing it on
bounds were established: a new row. We now have a structure like this:
2
From Wikipedia, the free encyclopedia Bell number
1
1 ’’x’’ # Set the first element of the current ro
# of the previous row
The value x here is determined by adding the number to current_row = [triangle[row_num-1][row_nu
the left of x (one) and the number above the number to
the left of x (also one). # Calculate the rest of the elements in t
# to the Bell triangle
1 1.upto(row_num) do |col_num|
1 2 sum = triangle[row_num-1][col_num-1]
y current_row.push(sum)
end
The value y is determined by copying over the number
from the right of the previous row. Since the number on
triangle[row_num] = current_row
the right hand side of the previous row has a value of 2, y
is given a value of two.
end
1
# Print out the Bell numbers
1 2
start.upto(finish) do |num|
2 3 ’’x’’
puts triangle[num-1][0]
Again, since x is not the leftmost element of a given row, end
its value is determined by taking the sum of the number end
to x’s left (three) and the number above the number to x’s
left (two). The sum is five. # Adjust the limits here
Here is the first five rows of this triangle: print_bell_numbers(1, 300)
1
1 2 And here an equivalent version written in Python
2 3 5
def bell_numbers(start, stop):
5 7 10 15
## Swap start and stop if start > stop
15 20 27 37 52
if stop = start:
Computer program yield t[-1][0] ## Yield the
row = [t[-1][-1]] ## Initializ
The following is example code in the Ruby programming
for b in t[-1]:
language that prints out all the Bell numbers from the 1st
row.append(row[-1] + b) ## Populate
to the 300th inclusive (the limits can be adjusted)
c += 1 ## We have f
t.append(row) ## Append th
#!/usr/bin/env ruby
for b in bell_numbers(1, 300):
def print_bell_numbers(start, finish)
print b
# Initialize the Bell triangle as a two-dimensional array
triangle = Array[Array[1]]
The number in the nth row and kth column is the number
of partitions of {1, ..., n} such that n is not together in one
# Make sure "start" is less than "finish", and both numbers are at least 1
class with any of the elements k, k + 1, ..., n − 1. For exam-
(finish, start = start, finish) if finish < start
ple, there are 7 partitions of {1, ..., 4} such that 4 is not
start = 1 if start < 1
together in one class with either of the elements 2, 3, and
finish = 1 if finish < 1
there are 10 partitions of {1, ..., 4} such that 4 is not to-
gether in one class with element 3. The difference is due
1.upto(finish-1) do |row_num|
to 3 partitions of {1, ..., 4} such that 4 is together in one
3
From Wikipedia, the free encyclopedia Bell number
class with element 2, but not with element 3. This corre-
sponds to the fact that there are 3 partitions of {1, ..., 3}
References
such that 3 is not together in one class with element 2: Spivey, Michael (2008), "A Generalized Recurrence
[1]
for counting partitions two elements which are always in 11,
for Bell Numbers", Journal of Integer Sequences 11
one class can be treated as just one element. The 3 ap- http://www.cs.uwaterloo.ca/journals/JIS/VOL11/
pears in the previous row of the table. Spivey/spivey25.pdf
• Gian-Carlo Rota (1964). "The Number of Partitions of
a Set". American Mathematical Monthly 71 (5): 498–504.
Prime Bell numbers doi:10.2307/2312585. MR0161805.
The first few Bell numbers that are primes are: • Lovász, L. (1993). Combinatorial Problems and Exercises
2, 5, 877, 27644437, (2nd ed. ed.). Amsterdam, Netherlands: North-
35742549198872617291353508656626642567, Holland.
359334085968622831041960188598043661065388726959079837 • Berend, D.; Tassa, T. (2010). "Improved Bounds on
Bell Numbers and on Moments of Sums of Random
corresponding to the indices 2, 3, 7, 13, 42 and 55 (se- Variables". Probability and Mathematical Statistics 30
quence A051130 in OEIS). (2): 185–205.
The next prime is B2841, which is approximately
9.30740105 × 106538. [1] As of 2006, it is the largest known
prime Bell number. Phil Carmody showed it was a prob- External links
able prime in 2002. After 17 months of computation with • Robert Dickau. "Diagrams of Bell numbers".
Marcel Martin’s ECPP program Primo, Ignacio Larrosa http://mathforum.org/advanced/robertd/
Cañestro proved it to be prime in 2004. He ruled out bell.html.
any other possible primes below B6000, later extended to • Pat Ballew. "Bell numbers".
B30447 by Eric Weisstein. http://www.pballew.net/Bellno.html.
• Weisstein, Eric W., "Bell Number" from MathWorld.
See also • Wagstaff, Samuel S. (1996). "Aurifeuillian
factorizations and the period of the Bell numbers
• Bell polynomials modulo a prime". Mathematics of computation 65 (213):
• Ordered Bell numbers – the number of weak orders 383–391. Bibcode 1996MaCom..65..383W. doi:10.1090/
• Stirling numbers of the second kind – the number of S0025-5718-96-00683-7. MR1325876.
ways to partition a set of n elements into k nonempty http://homes.cerias.purdue.edu/~ssw/bell/bell.ps.
subsets. • Gottfried Helms. "Further properties &
Generalization of Bell-Numbers". http://go.helms-
net.de/math/binomial/
04_5_SummingBellStirling.pdf.
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Categories:
• Integer sequences
• Triangles of numbers
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