Unsigned Stirling Number of the First Kind of n with n-1
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Theorem
Let $n \in \Z_{> 0}$ be an integer greater than $0$.
Then:
- $\ds {n \brack n - 1} = \binom n 2$
where:
- $\ds {n \brack n - 1}$ denotes an unsigned Stirling number of the first kind
- $\dbinom n 2$ denotes a binomial coefficient.
Proof
The proof proceeds by induction.
Basis for the Induction
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
- $\ds {n \brack n - 1} = \binom n 2$
$\map P 1$ is the case:
\(\ds {1 \brack 0}\) | \(=\) | \(\ds \delta_{1 0}\) | Unsigned Stirling Number of the First Kind of 1 | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Definition of Kronecker Delta | |||||||||||
\(\ds \) | \(=\) | \(\ds \binom 1 2\) | Definition of Binomial Coefficient |
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $\ds {k \brack k - 1} = \binom k 2$
from which it is to be shown that:
- $\ds {k + 1 \brack k} = \binom {k + 1} 2$
Induction Step
This is the induction step:
\(\ds {k + 1 \brack k}\) | \(=\) | \(\ds k {k \brack k} + {k \brack k - 1}\) | Definition of Unsigned Stirling Numbers of the First Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds k \times 1 + {k \brack k - 1}\) | Unsigned Stirling Number of the First Kind of Number with Self | |||||||||||
\(\ds \) | \(=\) | \(\ds k + \binom k 2\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds k + \frac {k \paren {k - 1} } 2\) | Binomial Coefficient with Two | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {k \paren {k - 1} + 2 k} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {k^2 - k + 2 k} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {k + 1} k} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \binom {k + 1} 2\) | Binomial Coefficient with Two |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \in \Z_{\ge 1}: {n \brack n - 1} = \dbinom n 2$
$\blacksquare$
Also see
- Signed Stirling Number of the First Kind of n with n-1
- Stirling Number of the Second Kind of n with n-1
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(49)$