Bell polynomials

Exponential Bell polynomials

The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by

B n , k ( x 1 , x 2 , … , x n − k + 1 ) = ∑ n ! j 1 ! j 2 ! ⋯ j n − k + 1 ! ( x 1 1 ! ) j 1 ( x 2 2 ! ) j 2 ⋯ ( x n − k + 1 ( n − k + 1 ) ! ) j n − k + 1 ,

where the sum is taken over all sequences j₁, j₂, j₃, ..., j_n−k+1 of non-negative integers such that these two conditions are satisfied:

j 1 + j 2 + ⋯ + j n − k + 1 = k , j 1 + 2 j 2 + 3 j 3 + ⋯ + ( n − k + 1 ) j n − k + 1 = n .

The sum

B n ( x 1 , … , x n ) = ∑ k = 1 n B n , k ( x 1 , x 2 , … , x n − k + 1 )

is called the nth complete exponential Bell polynomial.

Ordinary Bell polynomials

Likewise, the partial ordinary Bell polynomial, in contrast to the usual exponential Bell polynomial defined above, is given by

B ^ n , k ( x 1 , x 2 , … , x n − k + 1 ) = ∑ k ! j 1 ! j 2 ! ⋯ j n − k + 1 ! x 1 j 1 x 2 j 2 ⋯ x n − k + 1 j n − k + 1 ,

where the sum runs over all sequences j₁, j₂, j₃, ..., j_n−k+1 of non-negative integers such that

j 1 + j 2 + ⋯ + j n − k + 1 = k , j 1 + 2 j 2 + ⋯ + ( n − k + 1 ) j n − k + 1 = n .

The ordinary Bell polynomials can be expressed in the terms of exponential Bell polynomials:

B ^ n , k ( x 1 , x 2 , … , x n − k + 1 ) = k ! n ! B n , k ( 1 ! ⋅ x 1 , 2 ! ⋅ x 2 , … , ( n − k + 1 ) ! ⋅ x n − k + 1 ) .

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 {A, B, C}, it can be partitioned into two non-empty, non-overlapping subsets, which is also referred to as parts or blocks, in 3 different ways:

{{A}, {B, C}} {{B}, {A, C}} {{C}, {B, A}}

Thus, we can encode the information regarding these partitions as

B 3 , 2 ( x 1 , x 2 ) = 3 x 1 x 2 .

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, x₂ indicates the presence of a block with two elements. Similarly, x₁ indicates the presence of a block with a single element. The exponent of x_i^j indicates that there are j such blocks of size i in a single partition. Here, since both x₁ and x₂ has exponent 1, it indicates that there is only one such block in a given partition. The coefficient of the monomial 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 = x₁ⁿ.

As a more complicated example, consider

B 6 , 2 ( x 1 , x 2 , x 3 , x 4 , x 5 ) = 6 x 5 x 1 + 15 x 4 x 2 + 10 x 3 2 .

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.

Note that the sum of the subscripts in a monomials 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 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 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 the same by the total integer partition of n.

Also note that 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, j₁ + j₂ + ... = 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. 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 j₁ times, "2" appears j₂ 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

B 6 , 2 ( x 1 , x 2 , x 3 , x 4 , x 5 ) = 6 x 5 x 1 + 15 x 4 x 2 + 10 x 3 2

because there 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,

B 6 , 3 ( x 1 , x 2 , x 3 , x 4 ) = 15 x 4 x 1 2 + 60 x 3 x 2 x 1 + 15 x 2 3

because there 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.

Generating function

The exponential partial Bell polynomials can be defined by the double series expansion of its generating function:

Φ ( t , u ) = exp ⁡ ( u ∑ j = 1 ∞ x j t j j ! ) = ∑ n , k ≥ 0 B n , k ( x 1 , … , x n − k + 1 ) t n n ! u k = 1 + ∑ n = 1 ∞ t n n ! { ∑ k = 1 n u k B n , k ( x 1 , … , x n − k + 1 ) }

In other words, by what amounts to the same, by the series expansion of the exponential:

1 k ! ( ∑ j = 1 ∞ x j t j j ! ) k = ∑ n = k ∞ B n , k ( x 1 , … , x n − k + 1 ) t n n ! , k = 0 , 1 , 2 , …

The complete exponential Bell polynomial is defined by Φ ( t , 1 ) , or in other words:

Φ ( t , 1 ) = exp ⁡ ( ∑ j = 1 ∞ x j t j j ! ) = ∑ n = 0 ∞ B n ( x 1 , … , x n ) t n n ! .

Thus, the n-th complete Bell polynomial is given by

B n ( x 1 , … , x n ) = d n d t n exp ⁡ ( ∑ j = 1 ∞ x j t j j ! ) | t = 0 .

Likewise, the ordinary partial Bell polynomial can be defined by the generating function

Φ ^ ( t , u ) = exp ⁡ ( u ∑ j = 1 ∞ x j t j ) = ∑ k ≤ n B ^ n , k ( x 1 , … , x n − k + 1 ) t n u k k ! .

Or, equivalently, by series expansion of the exponential

( ∑ j = 1 ∞ x j t j ) k = ∑ n = k ∞ B ^ n , k ( x 1 , … , x n − k + 1 ) t n .

Recurrence relations

The complete Bell polynomials can be recursively defined as,

B n + 1 ( x 1 , … , x n + 1 ) = ∑ i = 0 n ( n i ) B n − i ( x 1 , … , x n − i ) x i + 1 ,

with the initialization B 0 = 1 .

The partial Bell polynomials can also be computed efficiently by a recursion relation

B n , k = ∑ i = 1 n − k + 1 ( n − 1 i − 1 ) x i B n − i , k − 1

where B 0 , 0 = 1 ; B n , 0 = 0 f o r n ≥ 1 ; B 0 , k = 0 f o r k ≥ 1.

The complete Bell polynomials also satisfy the binomial type relation

B n ( x 1 + y 1 , … , x n + y n ) = ∑ i = 0 n ( n i ) B n − i ( x 1 , … , x n − i ) B i ( y 1 , … , y i ) .

Determinant form

The complete Bell polynomial can be expressed as a determinant, as given by

B n ( x 1 , … , x n ) = det [ x 1 ( n − 1 1 ) x 2 ( n − 1 2 ) x 3 ( n − 1 3 ) x 4 ( n − 1 4 ) x 5 ⋯ ⋯ x n − 1 x 1 ( n − 2 1 ) x 2 ( n − 2 2 ) x 3 ( n − 2 3 ) x 4 ⋯ ⋯ x n − 1 0 − 1 x 1 ( n − 3 1 ) x 2 ( n − 3 2 ) x 3 ⋯ ⋯ x n − 2 0 0 − 1 x 1 ( n − 4 1 ) x 2 ⋯ ⋯ x n − 3 0 0 0 − 1 x 1 ⋯ ⋯ x n − 4 0 0 0 0 − 1 ⋯ ⋯ x n − 5 ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 0 0 0 ⋯ − 1 x 1 ] .

Stirling numbers and Bell numbers

The value of the Bell polynomial B_n,k(x₁,x₂,...) on the sequence of factorials equals an unsigned Stirling number of the first kind:

B n , k ( 0 ! , 1 ! , … , ( n − k ) ! ) = c ( n , k ) = | s ( n , k ) | = [ n k ] .

The value of the Bell polynomial B_n,k(x₁,x₂,...) on the sequence of ones equals a Stirling number of the second kind:

B n , k ( 1 , 1 , … , 1 ) = S ( n , k ) = { n k } .

The sum of these values gives the value of the complete Bell polynomial on the sequence of ones:

B n ( 1 , 1 , … , 1 ) = ∑ k = 1 n B n , k ( 1 , 1 , … , 1 ) = ∑ k = 1 n { n k } ,

which is the nth Bell number.

Inverse relations

If we define

y n = ∑ k = 1 n B n , k ( x 1 , … , x n − k + 1 ) ,

then we have the inverse relationship

x n = ∑ k = 1 n ( − 1 ) k − 1 ( k − 1 ) ! B n , k ( y 1 , … , y n − k + 1 ) .

Touchard polynomials

Touchard polynomial T n ( x ) = ∑ k = 0 n { n k } ⋅ x k can be expressed as the value of the complete Bell polynomial on all arguments being x:

T n ( x ) = B n ( x , x , … , x ) .

Convolution identity

For sequences x_n, y_n, n = 1, 2, ..., define a sort of convolution by:

( x ♢ y ) n = ∑ j = 1 n − 1 ( n j ) x j y n − j .

Note that the bounds of summation are 1 and n − 1, not 0 and n .

Let x n k ♢ be the nth term of the sequence

x ♢ ⋯ ♢ x ⏟ k   f a c t o r s .

Then

B n , k ( x 1 , … , x n − k + 1 ) = x n k ♢ k ! .

For example, let us compute B 4 , 3 ( x 1 , x 2 ) . We have

x = ( x 1   ,   x 2   ,   x 3   ,   x 4   , … ) x ♢ x = ( 0 ,   2 x 1 2   ,   6 x 1 x 2   ,   8 x 1 x 3 + 6 x 2 2   , … ) x ♢ x ♢ x = ( 0   ,   0   ,   6 x 1 3   ,   36 x 1 2 x 2   , … )

and thus,

B 4 , 3 ( x 1 , x 2 ) = ( x ♢ x ♢ x ) 4 3 ! = 6 x 1 2 x 2 .

Other identities

B n , k ( 1 ! , 2 ! , … , ( n − k + 1 ) ! ) = ( n − 1 k − 1 ) n ! k ! = L ( n , k ) which gives the Lah number

B n , k ( 1 , 2 , 3 , … , n − k + 1 ) = ( n k ) k n − k which gives the idempotent number

Examples

The first few complete Bell polynomials are:

B 0 = 1 , B 1 ( x 1 ) = x 1 , B 2 ( x 1 , x 2 ) = x 1 2 + x 2 , B 3 ( x 1 , x 2 , x 3 ) = x 1 3 + 3 x 1 x 2 + x 3 , B 4 ( x 1 , x 2 , x 3 , x 4 ) = x 1 4 + 6 x 1 2 x 2 + 4 x 1 x 3 + 3 x 2 2 + x 4 , B 5 ( x 1 , x 2 , x 3 , x 4 , x 5 ) = x 1 5 + 10 x 2 x 1 3 + 15 x 2 2 x 1 + 10 x 3 x 1 2 + 10 x 3 x 2 + 5 x 4 x 1 + x 5 B 6 ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = x 1 6 + 15 x 2 x 1 4 + 20 x 3 x 1 3 + 45 x 2 2 x 1 2 + 15 x 2 3 + 60 x 3 x 2 x 1 + 15 x 4 x 1 2 + 10 x 3 2 + 15 x 4 x 2 + 6 x 5 x 1 + x 6 , B 7 ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ) = x 1 7 + 21 x 1 5 x 2 + 35 x 1 4 x 3 + 105 x 1 3 x 2 2 + 35 x 1 3 x 4 + 210 x 1 2 x 2 x 3 + 105 x 1 x 2 3 + 21 x 1 2 x 5 + 105 x 1 x 2 x 4 + 70 x 1 x 3 2 + 105 x 2 2 x 3 + 7 x 1 x 6 + 21 x 2 x 5 + 35 x 3 x 4 + x 7 .

Faà di Bruno's formula

Faà di Bruno's formula may be stated in terms of Bell polynomials as follows:

d n d x n f ( g ( x ) ) = ∑ k = 1 n f ( k ) ( g ( x ) ) B n , k ( g ′ ( x ) , g ″ ( x ) , … , g ( n − k + 1 ) ( x ) ) .

Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose

f ( x ) = ∑ n = 1 ∞ a n n ! x n a n d g ( x ) = ∑ n = 1 ∞ b n n ! x n .

Then

g ( f ( x ) ) = ∑ n = 1 ∞ ∑ k = 1 n b k B n , k ( a 1 , … , a n − k + 1 ) n ! x n .

In particular, the complete Bell polynomials appear in the exponential of a formal power series:

exp ⁡ ( ∑ n = 1 ∞ a n n ! x n ) = ∑ n = 0 ∞ B n ( a 1 , … , a n ) n ! x n ,

which also represents the exponential generating function of the complete Bell polynomials on a fixed sequence of arguments a 1 , a 2 , … .

Reversion of series

Let two functions f and g be expressed in formal power series as

f ( w ) = ∑ k = 0 ∞ f k w k k ! , a n d g ( z ) = ∑ k = 0 ∞ g k z k k ! ,

such that g is the compositional inverse of f defined by g(f(w)) = w or f(g(z)) = z. If f₀ = 0 and f₁ ≠ 0, then an explicit form of the coefficients of the inverse can be given in term of Bell polynomials as

g n = 1 f 1 n ∑ k = 1 n − 1 ( − 1 ) k n k ¯ B n − 1 , k ( f ^ 1 , f ^ 2 , … , f ^ n − k ) , n ≥ 2 ,

with f ^ k = f k + 1 ( k + 1 ) f 1 , and n k ¯ = n ( n + 1 ) ⋯ ( n + k − 1 ) is the rising factorial, and g 1 = 1 f 1 .

Asymptotic expansion of Laplace-type integrals

Consider the integral of the form

I ( λ ) = ∫ a b e − λ f ( x ) g ( x ) d x ,

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 minima in [a,b] which occur at x=a. Assume that as x→a⁺,

f ( x ) ∼ f ( a ) + ∑ k = 0 ∞ a k ( x − a ) k + α , g ( x ) ∼ ∑ k = 0 ∞ b k ( x − a ) k + β − 1 ,

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

I ( λ ) ∼ e − λ f ( a ) ∑ n = 0 ∞ Γ ( n + β α ) c n λ ( n + β ) / α a s λ → ∞ ,

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:

c n = 1 α a 0 ( n + β ) / α ∑ k = 0 n b n − k ∑ j = 0 k ( − n + β α j ) 1 a 0 j B ^ k , j ( a 1 , a 2 , … , a k − j + 1 ) .

Symmetric polynomials

The elementary symmetric polynomial e n and the power sum symmetric polynomial p n can be related to each other using Bell polynomials as:

e n = ( − 1 ) n n ! B n ( s 1 , s 2 , … , s n ) ,                 w h e r e s l = − ( l − 1 ) ! p l , p n = ( − 1 ) n − 1 ( n − 1 ) ! ∑ k = 1 n ( − 1 ) k − 1 ( k − 1 ) ! B n , k ( e 1 , 2 ! e 2 , 3 ! e 3 , … , ( n − k + 1 ) ! e n − k + 1 ) .

This fact allows us to express the coefficients of monic polynomials in term the Bell polynomials of its roots. For instance, this can be applied with Cayley-Hamilton theorem to obtain the determinant of a n × n square matrix A in terms of its trace

det ( A ) = ( − 1 ) n n ! B n ( s 1 , s 2 , … , s n ) ,   w h e r e s l = − ( l − 1 ) ! t r ( A l ) .

Cycle index of symmetric groups

The cycle index of the symmetric group S n can be expressed in terms of complete Bell polynomials as follows:

Z ( S n ) = B n ( 0 ! a 1 , 1 ! a 2 , … , ( n − 1 ) ! a n ) n ! .

Moments and cumulants

The sum

μ n ′ = B n ( κ 1 , … , κ n ) = ∑ k = 1 n B n , k ( κ 1 , … , κ n − k + 1 )

is the nth raw moment of a probability distribution whose first n cumulants are κ₁, ..., κ_n. In other words, the nth moment is the nth complete Bell polynomial evaluated at the first n cumulants. Likewise, the nth cumulant can be given in terms of the moments as

κ n = ∑ k = 1 n ( − 1 ) k − 1 ( k − 1 ) ! B n , k ( μ 1 ′ , … , μ n − k + 1 ′ ) .

Hermite polynomials

The probabilists' Hermite polynomials can be expressed in terms of Bell polynomials as

H e n ( x ) = B n ( x , − 1 , 0 , … , 0 ) ,

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

exp ⁡ ( x t − t 2 2 ) = ∑ n = 0 ∞ H e n ( x ) t n n !

with that of Bell polynomials.

Representation of polynomial sequences of binomial type

For any sequence a₁, a₂, …, a_n of scalars, let

p n ( x ) = ∑ k = 1 n B n , k ( a 1 , … , a n − k + 1 ) x k .

Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity

p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) p n − k ( y ) . Example: For a₁ = … = a_n = 1, the polynomials p n ( x ) represent Touchard polynomials.

More generally, we have this result:

Theorem: All polynomial sequences of binomial type are of this form.

If we define a formal power series

h ( x ) = ∑ k = 1 ∞ a k k ! x k ,

then for all n,

h − 1 ( d d x ) p n ( x ) = n p n − 1 ( x ) .

Software

Bell polynomials are implemented in:

Mathematica as BellY

Maple as IncompleteBellB

SageMath as bell_polynomial