Stirling Number of the Second Kind of Number with Greater

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Theorem

Let $n, k \in \Z_{\ge 0}$.

Let $k > n$.


Let $\displaystyle \left\{ {n \atop k}\right\}$ denote a Stirling number of the second kind.

Then:

$\displaystyle \left\{ {n \atop k}\right\} = 0$


Proof 1

By definition, the Stirling numbers of the second kind are defined as the coefficients $\displaystyle \left\{ {n \atop k}\right\}$ which satisfy the equation:

$\displaystyle x^n = \sum_k \left\{ {n \atop k}\right\} x^{\underline k}$

where $x^{\underline k}$ denotes the $k$th falling factorial of $x$.


Both of the expressions on the left hand side and right hand side are polynomials in $x$ of degree $n$.

Hence the coefficient $\displaystyle \left\{ {n \atop k}\right\}$ of $x^{\underline k}$ where $k > n$ is $0$.

$\blacksquare$


Proof 2

The proof proceeds by induction.

For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:

$\displaystyle k > n \implies \left\{ {n \atop k}\right\} = 0$


Basis for the Induction

$P \left({0}\right)$ is the case:

$\displaystyle \left\{ {0 \atop k}\right\} = \delta_{0 k}$

from Stirling Number of the Second Kind of 0.


So by definition of Kronecker delta:

$\forall k \in \Z_{\ge 0}: k > 0 \implies \displaystyle \left\{ {0 \atop k}\right\} = 0$

and so $P \left({0}\right)$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that, if $P \left({r}\right)$ is true, where $0 \le r$, then it logically follows that $P \left({r + 1}\right)$ is true.


So this is the induction hypothesis:

$\displaystyle k > r \implies \left\{ {r \atop k}\right\} = 0$


from which it is to be shown that:

$\displaystyle k > r + 1 \implies \left\{ {r + 1 \atop k}\right\} = 0$


Induction Step

This is the induction step:


\(\displaystyle \left\{ {r + 1 \atop k}\right\}\) \(=\) \(\displaystyle k \left\{ {r \atop k}\right\} + \left\{ {r \atop k - 1}\right\}\)
\(\displaystyle \) \(=\) \(\displaystyle r \times 0 + \left\{ {r \atop k - 1}\right\}\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle r \times 0 + 0\) Induction Hypothesis: $k > r + 1 \implies k - 1 > r$
\(\displaystyle \) \(=\) \(\displaystyle 0\)

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\displaystyle \forall n \in \Z_{\ge 0}: k > n \implies \left\{ {n \atop k}\right\} = 0$

$\blacksquare$


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