gives the partial Bell polynomial .
gives the generalized partial Bell polynomial of a matrix m.
gives the generalized Bell polynomial of a matrix m.
- 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 mnn∧m1+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.
Examplesopen allclose all
Basic Examples (3)
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:
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)
Wolfram Research (2010), BellY, Wolfram Language function, https://reference.wolfram.com/language/ref/BellY.html.
Wolfram Language. 2010. "BellY." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BellY.html.
Wolfram Language. (2010). BellY. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BellY.html