matrixcalc (version 1.0-3)

stirling.matrix: 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

n

A positive integer value

Value

An order \(n + 1\) lower triangular matrix.

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 \(j \le i\). The numbers are also denoted by \(\left\{ {\begin{array}{*{20}{c}}i\\j\end{array}} \right\}\). Stirling numbers of the second kind can be computed recursively with the equation \(S_j^{i + 1} = S_{j - 1}^i + j\;S_j^i,\quad 1 \le i \le n - 1,\;1 \le j \le i\). The initial conditions for the recursion are \(S_i^i = 1,\quad 0 \le i \le n\) and \(S_j^0 = S_0^j = 0,\quad 0 \le j \le n\). The resultant numbers are organized in an order \(n + 1\) matrix \(\left[ {\begin{array}{*{20}{c}} {S_0^0}&0&0& \cdots &0\\ 0&{S_1^1}&0& \cdots &0\\ 0&{S_1^2}&{S_2^2}& \cdots &0\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0&{S_1^n}&{S_2^n}& \cdots &{S_n^n} \end{array}} \right]\).

References

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

Examples

# NOT RUN {
S <- stirling.matrix( 10 )
print( S )
# }