Stirling Matrix

Description

This function constructs and returns a Stirling matrix which is a lower triangular matrix containing the Stirling numbers of the second kind.

Usage

stirling.matrix(n)

Arguments

Details

The Stirling numbers of the second kind, S_i^j, are used in combinatorics to compute the number of ways a set of i objects can be partitioned into j non-empty subsets j ≤ i. The numbers are also denoted by ≤ft{ {egin{array}{*{20}{c}}i\jend{array}} ight}. Stirling numbers of the second kind can be computed recursively with the equation S_j^{i + 1} = S_{j - 1}^i + j;S_j^i,quad 1 ≤ i ≤ n - 1,;1 ≤ j ≤ i. The initial conditions for the recursion are S_i^i = 1,quad 0 ≤ i ≤ n and S_j^0 = S_0^j = 0,quad 0 ≤ j ≤ n. The resultant numbers are organized in an order n + 1 matrix ≤ft[ {egin{array}{*{20}{c}} {S_0^0}&0&0& cdots &0\ 0&{S_1^1}&0& cdots &0\ 0&{S_1^2}&{S_2^2}& cdots &0\ cdots & cdots & cdots & cdots & cdots \ 0&{S_1^n}&{S_2^n}& cdots &{S_n^n} end{array}} ight].

Value

An order n + 1 lower triangular matrix.

Note

If the argument n is not a positive integer, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Aceto, L. and D. Trigiante (2001). Matrices of Pascal and Other Greats, American Mathematical Monthly, March 2001, 108(3), 232-245.

Examples

S <- stirling.matrix( 10 )
print( S )

Results