# BellY

BellY[n,k,{x1,,xn-k+1}]

gives the partial Bell polynomial .

BellY[n,k,m]

gives the generalized partial Bell polynomial of a matrix m.

BellY[m]

gives the generalized Bell polynomial of a matrix m.

# Details • Mathematical function, suitable for both symbolic and numeric manipulations.
• The partial Bell polynomial can be used to express the  derivative of a composition of two functions through the Faà di Bruno formula .
• The BellY polynomial is given by   Boole[m1+2 m2++n mnnm1+m2++mnk] (xs/s!)ms.
• The generalized Bell polynomial can be used to express the  derivative of a composition of functions .
• BellY[n,k,m] is equivalent to BellY[ ] where is formed by adding UnitVector[n,k] to m as a first column.

# Examples

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## Basic Examples(3)

Partial Bell polynomial:

Generalized partial Bell polynomial:

Generalized Bell polynomial:

## Scope(1)

Evaluate for numerical matrix:

## Applications(8)

The generalized chain rule allows one to directly compute the  derivative of using BellY: . Verify this for symbolic and for low orders of :

For , this becomes the normal chain rule:

For , this is also known as Faà di Bruno's formula:

From the formula, it can be directly seen that the derivative is linear in with coefficients that are polynomial in , where is the polynomial coefficient of . Define a typesetting that makes this relation be obvious by paneling the Bell coefficients:

A few of the first derivatives:

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[n,z] as a special case:

## Properties & Relations(3)

Faà 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: