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
\(\left[ {\begin{array}{*{20}{c}}
{S_0^0}&0&0& \cdots &0\\
0&{S_1^1}&0& \cdots &0\\
0&{S_1^2}&{S_2^2}& \cdots &0\\
\cdots & \cdots & \cdots & \cdots & \cdots \\
0&{S_1^n}&{S_2^n}& \cdots &{S_n^n}
\end{array}} \right]\).