BUILT-IN MATHEMATICA SYMBOL

BellY

gives the partial Bell polynomial

.

BellY

gives the generalized partial Bell polynomial of a matrix m.

BellY[m]
gives the generalized Bell polynomial of a matrix m.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numeric manipulations.

The BellY polynomial is given by Boole[m₁+2 m₂++n m_nnm₁+m₂++m_nk] ()^m_s.

The partial Bell polynomial can be used to express the derivative of a composition of two functions through the Faa di Bruno formula .

The generalized Bell polynomial can be used to express the derivative of a composition of functions .

BellY is equivalent to BellY where is formed by adding UnitVector to m as a first column.

EXAMPLES

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Basic Examples (3)

Partial Bell polynomial:

Generalized partial Bell polynomial:

Generalized Bell polynomial:

Partial Bell polynomial:

In[1]:=

Out[1]=

Generalized partial Bell polynomial:

In[1]:=

Out[1]=

Generalized Bell polynomial:

In[1]:=

Out[1]=

Scope (1)

Evaluate for numerical matrix:

Generalizations & Extensions (1)

Find the term multiplying

in the expression for

:

Find the term directly from derivative:

Applications (7)

Compute the third raw moment in terms of cumulants:

Compute the third cumulant in terms of raw moments:

Compute a partial Bell polynomial using sum representation:

Compare with BellY:

Number of

-level labeled rooted trees with

leaves:

Alternatively:

Compute fourth-order derivatives of the Gamma function using the BellY polynomial:

Compute the series of an inverse function:

Find the number of ways to partition a set of 6 elements into two subsets from a partial Bell polynomial:

Check by explicit recursive generation of set partitions:

There are 10 ways to partition a set of 6 elements into two subsets of 3+3 elements:

There are 15 ways to partition a set of 6 elements into two subsets of 4+2 elements:

There are 6 ways to partition a set of 6 elements into two subsets of 5+1 elements:

Construct polynomial sequences of binomial type:

Verify their defining identity:

Recover BellB

as a special case:

Properties & Relations (3)

Faa di Bruno's formula for the third derivative of

:

Stirling numbers of the second kind:

Compute Bell numbers using generalized Bell polynomials:

Neat Examples (1)

Generate Bernoulli numbers using a generalized Bell polynomial: