Stirling Numbers Of The First Kind

The Stirling Numbers of the first kind, denoted $\begin{bmatrix}n\ k\end{bmatrix}$ or $s(n,k)$, are equivalently:

• The number of Permutations of $n$ elements with $k$ disjoint cycles.

• The $k$th coefficent in the expansions of the Falling Factorial $(x)_{n}$.

• The $k$th coefficient in the expansion of the Rising Factorial $x^{(n)}$.

• The numbers defined by the following Recurrance Relation:

  	$\displaystyle\begin{bmatrix}0\ 0\end{bmatrix}$	$\displaystyle=1$
  	$\displaystyle\begin{bmatrix}n+1\ 0\end{bmatrix}$	$\displaystyle=0$
  	$\displaystyle\begin{bmatrix}0\ k+1\end{bmatrix}$	$\displaystyle=0$
  	$\displaystyle\begin{bmatrix}n+1\ k+1\end{bmatrix}$	$\displaystyle=n\begin{bmatrix}n\ k+1\end{bmatrix}+\begin{bmatrix}n\ k\end{bmatrix}$