BellY

The partial (incomplete) Bell polynomial $B_{n,k}(x_1, x_2, \dots)$, by its textbook recurrence — built entirely from bigint binomial coefficients and compute-engine's own Add/Multiply, so it works the same for numeric or symbolic $x_i$.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageBellY✓↗

BellY(n, k, xs)$B_{n,k}(x_1, \dots, x_{n-k+1})$, from the list xs.

Typed signatures

Type	Package	Applies
(integer, integer, list) -> number	analytic	always

Domain: Special functions

Details

• Comtet's recurrence: $B_{n,k} = \sum_{i=1}^{n-k+1} \binom{n-1}{i-1} x_i B_{n-i,k-1}$, with $B_{0,0}=1$.
• All-ones arguments give the Stirling numbers of the second kind; $x_j = j!$ gives the (unsigned) Lah numbers — both checked directly against Wolfram's BellY.
• xs must have exactly $n-k+1$ elements, matching Wolfram's own arity requirement.

Examples

3 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{BellY}(4, 2, \bigl\lbrack\mathrm{x1}, \mathrm{x2}, \mathrm{x3}\bigr\rbrack)$Out$3\mathrm{x2}^2+4\mathrm{x1}\mathrm{x3}$

The partial Bell polynomial $B_{4,2}$ from three variables

live ↗

In$\mathrm{BellY}(4, 2, \bigl\lbrack x_1, x_2, x_3\bigr\rbrack)$Out$3x_2^2+4x_1x_3$

Applications #

Stays unexpanded for a sequence given symbolically with Array

live ↗

In$\mathrm{BellY}(6, 2, \mathrm{Array}(x, 6))$Out$\mathrm{BellY}(6, 2, \bigl\lbrack x(1), x(2), x(3), x(4), x(5), x(6)\bigr\rbrack)$

Properties #

All-ones arguments give Stirling(6, 2)

live ↗

In$\mathrm{BellY}(6, 2, \bigl\lbrack1, 1, 1, 1, 1\bigr\rbrack)$Out$31$

$x_j = j!$ gives the Lah number L(4, 2)

live ↗

In$\mathrm{BellY}(4, 2, \bigl\lbrack1, 2, 6\bigr\rbrack)$Out$36$

About

Examples	8
Checked against	Wolfram Language 8 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/bell-y.ts

See also: BellNumber, StirlingS2