some $c_1$ which is constant in ${\theta}$. Hence, ${\Delta W}$ has only real roots. Moreover, the only positive root of ${\Delta W}(N,k)$ occurs at $${\theta}= 1 \Big/ \left(\sum_{i=k+1}^{N-1} \frac{1}{i}\right).$$

We fix $N$ and argue by induction. The base case for $k = N-2$ matches the initial conditions in Corollary \[c:Wrec\] with $c_1(N,N-2) = (N-1)$.

Now suppose the result holds for $N-1$. Then, by Corollary \[c:Wrec\] $${\Delta W}(N,k) = (N-1) \left(  c_1 \left( \left(\sum_{i=k+1}^{N-2} \frac{1}{i}\right){\theta}- 1 \right) {\theta}{\{k\}!} \right) + {\theta}^2 \frac{(N-2)!}{k!} {\{k\}!}$$ $$= \left(  (N-1) c_1 \left( \left(\sum_{i=k+1}^{N-2} \frac{1}{i}\right){\theta}- 1 \right) + {\theta}\frac{(N-2)!}{k!} \right) {\theta}{\{k\}!}$$ $$= (N-1) c_1 \left( \left( \frac{(N-2)(N-3) \cdots (k+1)}{(N-1) c_1} + \frac{\sum\limits_{k+1 \leq i \leq N-2}\ \ \prod\limits_{\substack{k+1 \leq j \leq N-2 \\ j \neq i}} j }{(N-2)(N-3) \cdots (k+1)} \right) {\theta}- 1 \right) {\theta}{\{k\}!}.$$ Now, if we let $c_1 = (N-2)(N-3) \cdots (k+1)$, we may rewrite the linear term as $$\left( \frac{(N-2)(N-3) \cdots (k+1)}{(N-1)(N-2)(N-3) \cdots (k+1)} + \frac{(N-1)\sum\limits_{k+1 \leq i \leq N-2}\ \ \prod\limits_{\substack{k+1 \leq j \leq N-2 \\ j \neq i}} j }{(N-1)(N-2)(N-3) \cdots (k+1)} \right) {\theta}- 1  = \left( \sum_{i=k+1}^{N-1} \frac{1}{i}\right){\theta}- 1$$ obtaining $(N-1) c_1 \left( \left( \sum_{i=k+1}^{N-1} \frac{1}{i}\right){\theta}- 1
\right) {\theta}{\{k\}!}$ as desired.

\[c:ewensmain\] We have $${\kappa}_N({\theta}) = \begin{cases} 
        0 & \text{ 