Improving bounds on packing densities of 4-point permutations

Abstract

We consolidate what is currently known about packing densities of 4-pointpermutations and in the process improve the lower bounds for the packingdensities of 1324 and 1342. We also provide rigorous upper bounds for thepacking 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 afairly complete picture of all 4-point packing densities. We also provide newupper bounds for several small permutations of length at least five. Our maintool for the upper bounds is the framework of flag algebras introduced byRazborov in 2007.Comment: journal style, 18 page