copula (version 0.999-15)

Stirling: Eulerian and Stirling Numbers of First and Second Kind

Description

Compute Eulerian numbers and Stirling numbers of the first and second kind, possibly vectorized for all $k$ “at once”.

Usage

Stirling1(n, k)
Stirling2(n, k, method = c("lookup.or.store", "direct"))
Eulerian (n, k, method = c("lookup.or.store", "direct"))
Stirling1.all(n)
Stirling2.all(n)
Eulerian.all (n)

Arguments

n

positive integer (0 is allowed for Eulerian()).

k

integer in 0:n.

method

for Eulerian() and Stirling2(), string specifying the method to be used. "direct" uses the explicit formula (which may suffer from some cancelation for “large” n).

Value

$A(n,k)$, $s(n,k)$ or $S(n,k)$, respectively.Eulerian.all(n) is the same as sapply(0:(n-1), Eulerian, n=n) (for $n > 0$), Stirling1.all(n) is the same as sapply(1:n, Stirling1, n=n), and Stirling2.all(n) is the same as sapply(1:n, Stirling2, n=n), but more efficient.

Details

Eulerian numbers: $A(n,k) =$ the number of permutations of 1,2,...,n with exactly $k$ ascents (or exactly $k$ descents).

Stirling numbers of the first kind: s(n,k) = (-1)^n-k times the number of permutations of 1,2,...,n with exactly k cycles.

Stirling numbers of the second kind: $S(n,k)$ is the number of ways of partitioning a set of $n$ elements into $k$ non-empty subsets.

References

Eulerians:

NIST Digital Library of Mathematical Functions, 26.14: http://dlmf.nist.gov/26.14

Stirling numbers:

Abramowitz and Stegun 24,1,4 (p. 824-5 ; Table 24.4, p.835); Closed Form : p.824 "C."

NIST Digital Library of Mathematical Functions, 26.8: http://dlmf.nist.gov/26.8

Examples

Stirling1(7,2)
Stirling2(7,3)

Stirling1.all(9)
Stirling2.all(9)