Stirling Number of the Second Kind of n with n-2
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Theorem
Let $n \in \Z_{\ge 2}$ be an integer greater than or equal to $2$.
Then:
- $\ds {n \brace n - 2} = \binom {n + 1} 4 + 2 \binom n 4$
where:
- $\ds {n \brace n - 2}$ denotes an Stirling number of the second kind
- $\dbinom n 4$ denotes a binomial coefficient.
Proof
The proof proceeds by induction.
Basis for the Induction
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
- $\ds {n \brace n - 2} = \binom {n + 1} 4 + 2 \binom n 4$
$\map P 2$ is the case:
\(\ds {2 \brace 0}\) | \(=\) | \(\ds \delta_{2 0}\) | Definition of Stirling Numbers of the Second Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Definition of Kronecker Delta |
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $\ds {k \brace k - 2} = \binom {k + 1} 4 + 2 \binom k 4$
from which it is to be shown that:
- $\ds {k + 1 \brace k - 1} = \binom {k + 2} 4 + 2 \binom {k + 1} 4$
Induction Step
This is the induction step:
\(\ds {k + 1 \brace k - 1}\) | \(=\) | \(\ds \paren {k - 1} {k \brace k - 1} + {k \brace k - 2}\) | Definition of Stirling Numbers of the Second Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k - 1} \binom k 2 + {k \brace k - 2}\) | Stirling Number of the Second Kind of n with n-1 | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k - 1} \binom k 2 + \binom {k + 1} 4 + 2 \binom k 4\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {k - 1} \frac {k \paren {k - 1} } 2 + \dfrac {\paren {k + 1} k \paren {k - 1} \paren {k - 2} } {4!} + 2 \dfrac {k \paren {k - 1} \paren {k - 2} \paren {k - 3} } {4!}\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds k \paren {k - 1} \paren {\frac {k - 1} 2 + \dfrac {\paren {k + 1} \paren {k - 2} } {4!} + 2 \dfrac {\paren {k - 2} \paren {k - 3} } {4!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k \paren {k - 1} \paren {\frac {12 k - 12 + k^2 - k - 2 + 2 k^2 - 10 k + 12} {4!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k \paren {k - 1} \paren {\frac {k^2 + 3 k + 2 + 2 k^2 - 2 k + 4} {4!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k \paren {k - 1} \paren {\frac {\paren {k + 2} \paren {k + 1} + 2 \paren {k + 1} \paren {k - 2} } {4!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {k + 2} \paren {k + 1} k \paren {k - 1} } {4!} + 2 \frac {\paren {k + 1} k \paren {k - 1} \paren {k - 2} } {4!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \binom {k + 2} 4 + 2 \binom {k + 1} 4\) | Definition of Binomial Coefficient |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \in \Z_{\ge 2}: {n \brace n - 2} = \binom {n + 1} 4 + 2 \binom n 4$
$\blacksquare$
Also see
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(57)$