Given $n$ point masses on the real axis with their initial positions and velocities, determine the total number of collisions (from $t=0$ to $t=\infty$). Here I suppose that the collisions are elastic, i.e the momentum and energy of 2 particles just before and after each collision are conserved.
For example, if all particles have the same mass, then after each collision, 2 particles exchange their velocities, so we can imagine 2 particles pass through each other. Let $\sigma_0 = id$ and $\sigma_T$ the order of particles for $t=0$ and $t=T$, where $T$ sufficiently large so that there is no collision for $t>T$. The the number of collision is the inversion number of $\sigma_T$
What about the general case? What if we replace points by segments? Could you give me some results one has obtained so far?