This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# BellY

 BellYgives the partial Bell polynomial . BellYgives the generalized partial Bell polynomial of a matrix m. BellY[m]gives the generalized Bell polynomial of a matrix m.
• Mathematical function, suitable for both symbolic and numeric manipulations.
• The BellY polynomial is given by Boole[m1+2 m2++n mnnm1+m2++mnk] ()ms.
• 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 .
Partial Bell polynomial:
Generalized partial Bell polynomial:
Generalized Bell polynomial:
Partial Bell polynomial:
 Out[1]=

Generalized partial Bell polynomial:
 Out[1]=

Generalized Bell polynomial:
 Out[1]=
 Scope   (1)
Evaluate for numerical matrix:
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:
Faa di Bruno's formula for the third derivative of :
Stirling numbers of the second kind:
Compute Bell numbers using generalized Bell polynomials:
Generate Bernoulli numbers using a generalized Bell polynomial:
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