For given $N$, $x$ and $k$ such that $0\leq x
\begin{align} & \frac{(N+1-p)(N-p)}{2}\leq \frac{(N+1-x-k)(N-x-k)}{2}+\frac{(x+1)x}{2}<\frac{(N+1-p)(N-p)}{2}+\frac{p(p-1)}{2}+p+1-x \end{align}
For given $N$, $x$ and $k$ such that $0\leq x
\begin{align} & \frac{(N+1-p)(N-p)}{2}\leq \frac{(N+1-x-k)(N-x-k)}{2}+\frac{(x+1)x}{2}<\frac{(N+1-p)(N-p)}{2}+\frac{p(p-1)}{2}+p+1-x \end{align}