Consider $n$ line segments in the Cartesian plane. For $1\leq k\leq n$, the $k$-th line segment is drawn from $(k,0)$ to $(x_k,1)$, where $\{x_1,x_2,...,x_k\}$ is a permutation of $\{1,2,...,n\}$. Define $a(n)$ to be the maximal number of distinct points at which two or more line segments intersect.
For example, take $n=3$. A maximal construction for $a(n)$ would be a segment from $(1,0)$ to $(3,1)$, $(2,0)$ to $(1,1)$, and $(3,0)$ to $(2,1)$, giving $2$ distinct intersection points.
I have simulated the maximal values up to $n=13$ (and my friend has posted it on OEIS already here: https://oeis.org/A332774), but I am limited by the extreme time complexity of my program (around $n*n!*\log n$). The values from $n=1$ to $16$ are ($n=14$ to $16$ provided by Misha Lavrov):
$0,1,2,5,8,13,17,23,30,39,47,57,67,79,90,103$
I was wondering if a formula for $a(n)$ could exist, either recursive or explicit. Could it have something to do with the number of intersections in certain types of bipartite graphs? If no formula can be found, can there be any good estimates for upper / lower bounds?
Edit: Misha Lavrov has commented that there could be a repeating pattern of 8 in the second differences, though we do not have significant data supporting this claim.