# pr.probability: Probability that the perfect match is not nested on the N route with the given interval?

Given a trajectory graph $$P_n$$ ($$n$$ even). We add new set of $$n / 2$$ borders $$C$$ between non-adjacent path nodes such that establish $$C$$ It forms a perfect combination. The output graph is the union of the route. $$P_n$$ and perfect match $$C$$. Each edge $$e$$ connects two non-adjacent nodes $$V_i$$ Y $$V_j$$ in the path $$P_n$$. The lapse of an edge $$e (i, j)$$ in perfect match it's the same $$abs (i-j)$$. The perfect match span $$C$$ is $$max_ {e in C} span (e)$$ .

What is the probability that there is a perfect match without nesting when the perfect match interval is at most? $$n ^ r$$ where $$r lt 1$$?

This was motivated by this post.