Bounding quantities related to the packing density of 1(L+1)L...2

We bound several quantities related to the packing density of the patterns 1(L+1)L…2. These bounds sharpen results of B'ona, Sagan, and Vatter and give a new proof of the packing density of these patterns, originally computed by Stromquist in the case L=2 and by Price for larger L. We end with comments and conjectures.