Of the ten 3-element subsequences of the sequence s = 13254, four (132, 154, 354, and 254) are order-isomorphic to the pattern 132. We say that the "packing density" of 132 in s is 4/10. In this talk we consider various patterns and find sequences s of various lengths that maximize the packing density. We examine especially the limiting case for long sequences s. We reveal the "averaging method" for proving the existence of limits, define "layered" permutations, and make a connection to partially ordered sets.