◀ ▲ ▶Branches / Combinatorics / Explanation: Combinatorial Interpretation of Stirling Numbers of the First Kind
The following interpretation of the Stirling number of first type is sometimes in the literature used for their definition:# Explanation: Combinatorial Interpretation of Stirling Numbers of the First Kind
(related to Part: Stirling Numbers)
Let $n,r\ge 0$ be integers. The Stirling number of the first type $\left[\begin{array}{c}n\\r\end{array}\right]$ can be interpreted as the number of ways to arrange $n$ objects into $r$ cycles. In other words, the Stirling number of first type $\left[\begin{array}{c}n\\r\end{array}\right]$ is the number of permutations of $n$ objects with exactly $r$ cycles.
Examples
- $\left[\begin{array}{c}4\\2\end{array}\right]=11,$ since there are $11$ ways to arrange $4$ objects into $2$ cycles: $$\begin{array}{ccc}
[a,b,c][d]&[a,b,d][c]&[a,c,d][b]\\
[b,c,d][a]&[a,c,b][d]&[a,d,b][c]\\
[a,d,c][b]&[b,d,c][a]\\
[a,b][c,d]&[a,c],[b,d]&[a,d][b,c]
\end{array}$$
- $\left[\begin{array}{c}n\\1\end{array}\right]=(n-1)!,$ since in every cycle $[a_1,a_2\ldots,a_n]$ of the length $n$ and a fixed element $a_1$ we can arrange $a_2,\ldots,a_n$ in the factorial $(n-1)!$ ways.
Convention
$\left[\begin{array}{c}n\\r\end{array}\right]$ can be verbalized as "$n$ cycle $r$."
Mentioned in:
Proofs: 1
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References
Bibliography
- Aigner, Martin: "Diskrete Mathematik", vieweg studium, 1993
