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BellY—Wolfram Documentation

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BellY[n,k,{x₁,…,x_n-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.

Partial Bell polynomial:

Generalized partial Bell polynomial:

Generalized Bell polynomial:

Evaluate for a numerical matrix:

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:

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:

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:

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:

Generate Bernoulli numbers using a generalized Bell polynomial:

Generate Euler numbers using a generalized Bell polynomial:

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