In
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
mathematics, the Bell polynomials, named in honor of
Eric Temple Bell
Eric Temple Bell (7 February 1883 – 21 December 1960) was a Scottish-born mathematician and science fiction writer who lived in the United States for most of his life. He published non-fiction using his given name and fiction as John Tai ...
, are used in the study of set partitions. They are related to
Stirling
Stirling (; sco, Stirlin; gd, Sruighlea ) is a city in central Scotland, northeast of Glasgow and north-west of Edinburgh. The market town, surrounded by rich farmland, grew up connecting the royal citadel, the medieval old town with its me ...
and
Bell numbers. They also occur in many applications, such as in the
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...
.
Definitions
Exponential Bell polynomials
The ''partial'' or ''incomplete'' exponential Bell polynomials are a
triangular array
In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the ''i''th row contains only ''i'' elements.
Examples
Notable ...
of polynomials given by
:
where the sum is taken over all sequences ''j''
1, ''j''
2, ''j''
3, ..., ''j''
''n''−''k''+1 of non-negative integers such that these two conditions are satisfied:
:
:
The sum
:
is called the ''n''th ''complete exponential Bell polynomial''.
Ordinary Bell polynomials
Likewise, the partial ''ordinary'' Bell polynomial is defined by
:
where the sum runs over all sequences ''j''
1, ''j''
2, ''j''
3, ..., ''j''
''n''−''k''+1 of non-negative integers such that
:
:
The ordinary Bell polynomials can be expressed in the terms of exponential Bell polynomials:
:
In general, Bell polynomial refers to the exponential Bell polynomial, unless otherwise explicitly stated.
Combinatorial meaning
The exponential Bell polynomial encodes the information related to the ways a set can be partitioned. For example, if we consider a set , it can be partitioned into two non-empty, non-overlapping subsets, which is also referred to as parts or blocks, in 3 different ways:
:
:
:
Thus, we can encode the information regarding these partitions as
:
Here, the subscripts of ''B''
3,2 tells us that we are considering the partitioning of set with 3 elements into 2 blocks. The subscript of each ''x''
i indicates the presence of block with ''i'' elements (or block of size ''i'') in a given partition. So here, ''x''
2 indicates the presence of a block with two elements. Similarly, ''x''
1 indicates the presence of a block with a single element. The exponent of ''x''
ij indicates that there are ''j'' such blocks of size ''i'' in a single partition. Here, since both ''x''
1 and ''x''
2 has exponent 1, it indicates that there is only one such block in a given partition. The coefficient of the
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
indicates how many such partitions there are. For our case, there are 3 partitions of a set with 3 elements into 2 blocks, where in each partition the elements are divided into two blocks of sizes 1 and 2.
Since any set can be divided into a single block in only one way, the above interpretation would mean that ''B''
''n'',1 = ''x''
''n''. Similarly, since there is only one way that a set with ''n'' elements be divided into ''n'' singletons, ''B''
''n'',''n'' = ''x''
1''n''.
As a more complicated example, consider
:
This tells us that if a set with 6 elements is divided into 2 blocks, then we can have 6 partitions with blocks of size 1 and 5, 15 partitions with blocks of size 4 and 2, and 10 partitions with 2 blocks of size 3.
The sum of the subscripts in a monomial is equal to the total number of elements. Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer ''n'' can be expressed as a summation of ''k'' positive integers. This is the same as the
integer partition
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same part ...
of ''n'' into ''k'' parts. For instance, in the above examples, the integer 3 can be partitioned into two parts as 2+1 only. Thus, there is only one monomial in ''B''
3,2. However, the integer 6 can be partitioned into two parts as 5+1, 4+2, and 3+3. Thus, there are three monomials in ''B''
6,2. Indeed, the subscripts of the variables in a monomial are the same as those given by the integer partition, indicating the sizes of the different blocks. The total number of monomials appearing in a complete Bell polynomial ''B
n'' is thus equal to the total number of integer partitions of ''n''.
Also the degree of each monomial, which is the sum of the exponents of each variable in the monomial, is equal to the number of blocks the set is divided into. That is, ''j''
1 + ''j''
2 + ... = ''k'' . Thus, given a complete Bell polynomial ''B
n'', we can separate the partial Bell polynomial ''B
n,k'' by collecting all those monomials with degree ''k''.
Finally, if we disregard the sizes of the blocks and put all ''x''
''i'' = ''x'', then the summation of the coefficients of the partial Bell polynomial ''B''
''n'',''k'' will give the total number of ways that a set with ''n'' elements can be partitioned into ''k'' blocks, which is the same as the
Stirling numbers of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \le ...
. Also, the summation of all the coefficients of the complete Bell polynomial ''B
n'' will give us the total number of ways a set with ''n'' elements can be partitioned into non-overlapping subsets, which is the same as the Bell number.
In general, if the integer ''n'' is
partitioned into a sum in which "1" appears ''j''
1 times, "2" appears ''j''
2 times, and so on, then the number of
partitions of a set of size ''n'' that collapse to that partition of the integer ''n'' when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.
Examples
For example, we have
:
because the ways to partition a set of 6 elements as 2 blocks are
:6 ways to partition a set of 6 as 5 + 1,
:15 ways to partition a set of 6 as 4 + 2, and
:10 ways to partition a set of 6 as 3 + 3.
Similarly,
:
because the ways to partition a set of 6 elements as 3 blocks are
:15 ways to partition a set of 6 as 4 + 1 + 1,
:60 ways to partition a set of 6 as 3 + 2 + 1, and
:15 ways to partition a set of 6 as 2 + 2 + 2.
Properties
Generating function
The exponential partial Bell polynomials can be defined by the double series expansion of its generating function:
:
In other words, by what amounts to the same, by the series expansion of the ''k''-th power:
:
The complete exponential Bell polynomial is defined by
, or in other words:
:
Thus, the ''n''-th complete Bell polynomial is given by
:
Likewise, the ''ordinary'' partial Bell polynomial can be defined by the generating function
:
Or, equivalently, by series expansion of the ''k''-th power:
:
See also
generating function transformations for Bell polynomial generating function expansions of compositions of sequence
generating functions and
powers
Powers may refer to:
Arts and media
* ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming
** ''Powers'' (American TV series), a 2015–2016 series based on the comics
* ''Powers'' (British TV series), a 200 ...
,
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s, and
exponentials
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
* Exp ...
of a sequence generating function. Each of these formulas is cited in the respective sections of Comtet.
Recurrence relations
The complete Bell polynomials can be
recurrently defined as
:
with the initial value
.
The partial Bell polynomials can also be computed efficiently by a recurrence relation:
:
where
:
:
:
The complete Bell polynomials also satisfy the following recurrence differential formula:
:
Derivatives
The partial derivatives of the complete Bell polynomials are given by
:
Similarly, the partial derivatives of the partial Bell polynomials are given by
:
If the arguments of the Bell polynomials are one-dimensional functions, the chain rule can be used to obtain
:
Determinant forms
The complete Bell polynomial can be expressed as
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
s:
:
and
:
Stirling numbers and Bell numbers
The value of the Bell polynomial ''B''
''n'',''k''(''x''
1,''x''
2,...) on the sequence of
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) ...
s equals an unsigned
Stirling number of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed poin ...
:
:
The sum of these values gives the value of the complete Bell polynomial on the sequence of factorials:
:
The value of the Bell polynomial ''B''
''n'',''k''(''x''
1,''x''
2,...) on the sequence of ones equals a
Stirling number of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \le ...
:
:
The sum of these values gives the value of the complete Bell polynomial on the sequence of ones:
:
which is the ''n''th
Bell number
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of epony ...
.
Inverse relations
If we define
:
then we have the inverse relationship
:
Touchard polynomials
Touchard polynomial
can be expressed as the value of the complete Bell polynomial on all arguments being ''x'':
:
Convolution identity
For sequences ''x''
''n'', ''y''
''n'', ''n'' = 1, 2, ..., define a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
by:
:
The bounds of summation are 1 and ''n'' − 1, not 0 and ''n'' .
Let
be the ''n''th term of the sequence
:
Then
:
For example, let us compute
. We have
:
:
:
and thus,
:
Other identities
*
which gives the
Lah number.
*
which gives the
idempotent number.
*
and
.
* The complete Bell polynomials satisfy the binomial type relation:
*:
*:
:This corrects the omission of the factor
in Comtet's book.
* When
,
:
* Special cases of partial Bell polynomials:
:
Examples
The first few complete Bell polynomials are:
:
Applications
Faà di Bruno's formula
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...
may be stated in terms of Bell polynomials as follows:
:
Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose
:
Then
:
In particular, the complete Bell polynomials appear in the exponential of a
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
:
:
which also represents the
exponential generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
of the complete Bell polynomials on a fixed sequence of arguments
.
Reversion of series
Let two functions ''f'' and ''g'' be expressed in formal power series as
:
such that ''g'' is the compositional inverse of ''f'' defined by ''g''(''f''(''w'')) = ''w'' or ''f''(''g''(''z'')) = ''z''. If ''f''
0 = 0 and ''f''
1 ≠ 0, then an explicit form of the coefficients of the inverse can be given in term of Bell polynomials as
:
with
and
is the rising factorial, and
Asymptotic expansion of Laplace-type integrals
Consider the integral of the form
:
where (''a'',''b'') is a real (finite or infinite) interval, λ is a large positive parameter and the functions ''f'' and ''g'' are continuous. Let ''f'' have a single minimum in
'a'',''b''which occurs at ''x'' = ''a''. Assume that as ''x'' → ''a''
+,
:
:
with ''α'' > 0, Re(''β'') > 0; and that the expansion of ''f'' can be term wise differentiated. Then, Laplace–Erdelyi theorem states that the asymptotic expansion of the integral ''I''(''λ'') is given by
:
where the coefficients ''c
n'' are expressible in terms of ''a
n'' and ''b
n'' using partial ''ordinary'' Bell polynomials, as given by Campbell–Froman–Walles–Wojdylo formula:
:
Symmetric polynomials
The
elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary s ...
and the
power sum symmetric polynomial In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a su ...
can be related to each other using Bell polynomials as:
:
:
These formulae allow one to express the coefficients of monic polynomials in terms of the Bell polynomials of its zeroes. For instance, together with
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...
they lead to expression of the determinant of a ''n'' × ''n'' square matrix ''A'' in terms of the traces of its powers:
:
Cycle index of symmetric groups
The
cycle index
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in ...
of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
can be expressed in terms of complete Bell polynomials as follows:
:
Moments and cumulants
The sum
:
is the ''n''th raw
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
of a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
whose first ''n''
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will hav ...
s are ''κ''
1, ..., ''κ''
''n''. In other words, the ''n''th moment is the ''n''th complete Bell polynomial evaluated at the first ''n'' cumulants. Likewise, the ''n''th cumulant can be given in terms of the moments as
:
Hermite polynomials
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well ...
can be expressed in terms of Bell polynomials as
:
where ''x''
''i'' = 0 for all ''i'' > 2; thus allowing for a combinatorial interpretation of the coefficients of the Hermite polynomials. This can be seen by comparing the generating function of the Hermite polynomials
:
with that of Bell polynomials.
Representation of polynomial sequences of binomial type
For any sequence ''a''
1, ''a''
2, …, ''a''
''n'' of scalars, let
:
Then this polynomial sequence is of
binomial type, i.e. it satisfies the binomial identity
:
:Example: For ''a''
1 = … = ''a''
''n'' = 1, the polynomials
represent
Touchard polynomials
The Touchard polynomials, studied by , also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by
:T_n(x)=\sum_^n S(n,k)x^k=\sum_^n
\left\x^k,
where S(n,k)=\left\is a Stirling numb ...
.
More generally, we have this result:
:Theorem: All polynomial sequences of binomial type are of this form.
If we define a formal power series
:
then for all ''n'',
:
Software
Bell polynomials are implemented in:
*
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimi ...
a
BellY*
Maple
''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since ht ...
a
IncompleteBellB*
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, nu ...
a
bell_polynomial
See also
*
Bell matrix In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone ...
*
Exponential formula
Notes
References
*
*
*
*
*
* (contains also elementary review of the concept Bell-polynomials)
*
*
*
*
*
*
*
*
{{DEFAULTSORT:Bell Polynomials
Enumerative combinatorics
Polynomials