| stirling.matrix {matrixcalc} | R Documentation |
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 |
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}{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\lbrack {\begin{array}{ccccc}
{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\rbrack.
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 )