Improving bounds on packing densities of 4-point permutations.

We consolidate what is currently known about packing densities of 4-point permutations and in the process improve the lower bounds for the packing densities of 1324 and 1342. We also provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within $10^{-4}$ of the true packing densities. Together with the known bounds, this gives us a fairly complete picture of all 4-point packing densities. We also provide new upper bounds for several small permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007.