multicool (version 1.0.1)

Stirling2: Compute Stirling numbers of the second kind

Description

This function computes Stirling numbers of the second kind, $S(n,k)$, which count the number of ways of partitioning n distinct objects in to k non-empty sets.

Usage

Stirling2(n, k)
S2(n, k)

Value

An vector of Stirling numbers of the second kind

Arguments

n

A vector of one or more positive integers

k

A vector of one or more positive integers

Functions

• S2(): Compute Stirling numbers of the second kind

Author

James Curran

Details

The implementation on this function is a simple recurrence relation which defines $$S(n,k)=kS(n-1,k),+S(n-1,k-1)$$ for $k>0$ with the inital conditions $S(0,0)=1$ and $S(n,0)=S(0,n)=0$. If n and n have different lengths then expand.grid is used to construct a vector of (n, k) pairs

References

https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Recurrence_relation

Examples

## returns S(6, 3)
Stirling2(6, 3)

## returns S(6,1), S(6,2), ..., S(6,6)
S2(6, 1:6)

## returns S(6,1), S(5, 2), S(4, 3)
S2(6:4, 1:3)