Question 162
Let $n \geqslant 3$ be an integer. A sequence $P_{1}, P_{2}, \ldots, P_{n}$ of distinct points in the plane is called good if no three of them are collinear, the polyline $P_{1} P_{2} \ldots P_{n}$ is non-self-intersecting and the triangle $P_{i} P_{i+1} P_{i+2}$ is oriented counterclockwise for every $i=1,2, \ldots, n-2$.
For every integer $n \geqslant 3$ determine the greatest possible integer $k$ with the following property: there exist $n$ distinct points $A_{1}, A_{2}, \ldots, A_{n}$ in the plane for which there are $k$ distinct permutations $\sigma:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)}$ is good.
(A polyline $P_{1} P_{2} \ldots P_{n}$ consists of the segments $P_{1} P_{2}, P_{2} P_{3}, \ldots, P_{n-1} P_{n}$.)
For every integer $n \geqslant 3$ determine the greatest possible integer $k$ with the following property: there exist $n$ distinct points $A_{1}, A_{2}, \ldots, A_{n}$ in the plane for which there are $k$ distinct permutations $\sigma:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)}$ is good.
(A polyline $P_{1} P_{2} \ldots P_{n}$ consists of the segments $P_{1} P_{2}, P_{2} P_{3}, \ldots, P_{n-1} P_{n}$.)