nStirling2 {kStatistics}

Stirling number of second kind

Description

The function computes the Stirling number of the second kind.

Usage

nStirling2( n, k ) 

Arguments

n

integer

k

integer less or equal to n

Details

The Stirling number of the second kind S(n,k) is equal to the number of ways to split a set of cardinality n into k nonempty subsets. For example, if the set is [a,b,c,d], then the partitions in 2 blocks are: [[a], [bcd]], [[b], [acd]], [[c], [abd]], [[d],[abc]] with cardinalities (1,3) and [ab, cd], [ac, bd], [ad, bc] with cardinalities (2,2). Then S(4,2) is equal to 7. Note that (1,3) and (2,2) are also the partitions of the integer 4 in 2 parts.

Value

integer

the Stirling number of the second kind

Note

Called by the nKS and nKM functions in the kStatistics package.

Author(s)

Elvira Di Nardo [e-mail],
Giuseppe Guarino [e-mail]

References

R. P. Stanley (2012) Enumerative combinatorics. Vol.1. II edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge.

See Also

mkmSet, mCoeff, intPart, countP, ff

Examples

# Return the number of ways to split a set of 6 objects into 2 nonempty subsets
nStirling2(6,2)