Definition: An edge-component is a sequence of some consecutive collinear-segments.
Consider an $n \times n$ grid-like arrangement of $2n$ lines. Is there any idea about the number of simple cycles with exactly $2n$ edge-components such that each line in the grid contains exactly one edge-component of the cycle? In other words, the cycles should traverse all the lines. A simple cycle is a cycle which passes through the nodes exactly once.
In the picture you can see a $7\times7$ grid-like arrangement and a desired 14-cycle.
Any hint or observation is helpful.