# 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 prepending 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 a numerical matrix:

## Applications(13)

### Calculus(3)

The generalized chain rule allows one to directly compute the derivative of using BellY: . Verify this with 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 rule that makes this relation obvious by paneling the Bell coefficients:

A few of the first derivatives:

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

Compare with the explicit evaluation of the derivative:

Compute the series of an inverse function:

Compare with the result of InverseSeries:

### Combinatorics(6)

Compute Stirling numbers of the first kind in terms of the partial Bell polynomials:

Compute Stirling numbers of the second kind in terms of the partial Bell polynomials:

Compute Bell numbers using generalized Bell polynomials:

Compute Bell polynomials BellB[n,z] using generalized Bell polynomials:

Compute Catalan numbers using generalized Bell polynomials:

Number of -level labeled rooted trees with leaves:

Compare with an alternative formula:

The cycle index polynomial of the symmetric group of degree n:

Compare with the result of CycleIndexPolynomial:

The cycle index polynomial of the alternating group of degree n:

Compare with the result of CycleIndexPolynomial:

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:

### Other Applications(4)

Define the complete Bell polynomial of n variables:

Show the first few complete Bell polynomials:

Compute the third raw moment in terms of cumulants:

Compute the third cumulant in terms of raw moments:

Construct polynomial sequences of binomial type:

Verify their defining identity:

Recover BellB[n,z] as a special case:

The n elementary symmetric polynomial can be defined in terms of BellY:

Compare with SymmetricPolynomial for the case of five variables:

## Properties & Relations(6)

Compute a partial Bell polynomial using its sum representation:

Compare with BellY:

Compute a partial Bell polynomial using Cvijović's iterated sum formula:

Compare with BellY:

A linear combination of partial Bell polynomials:

The equivalent expression in terms of the generalized Bell polynomial:

A generalized partial Bell polynomial of a matrix:

This can be computed in terms of the generalized Bell polynomial by prepending a unit vector as a column:

A linear combination of generalized partial Bell polynomials of a matrix can be expressed as a generalized Bell polynomial by prepending the column of coefficients to the matrix:

Faà di Bruno's formula for the third derivative of :

Demonstrate an inversion relation for generalized Bell polynomials:

## Neat Examples(2)

Generate Bernoulli numbers using a generalized Bell polynomial:

Generate Euler numbers using a generalized Bell polynomial:

Wolfram Research (2010), BellY, Wolfram Language function, https://reference.wolfram.com/language/ref/BellY.html.

#### Text

Wolfram Research (2010), BellY, Wolfram Language function, https://reference.wolfram.com/language/ref/BellY.html.

#### CMS

Wolfram Language. 2010. "BellY." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BellY.html.

#### APA

Wolfram Language. (2010). BellY. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BellY.html

#### BibTeX

@misc{reference.wolfram_2024_belly, author="Wolfram Research", title="{BellY}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BellY.html}", note=[Accessed: 25-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_belly, organization={Wolfram Research}, title={BellY}, year={2010}, url={https://reference.wolfram.com/language/ref/BellY.html}, note=[Accessed: 25-July-2024 ]}