The Packing Density of Other Layered Permutations

In this paper the packing density of various layered permutations is calculated, thus solving some problems suggested by Albert, Atkinson, Handley, Holton & Stromquist [Electron. J. Combin. 9 (2002), #R5]. Specifically, the density is found for layered permutations of type [m1, . . . ,mr] when log(r +1) ≤ min{mi}. It is also shown how to derive good estimates for the packing density of permutations of type [k, 1, k] when k ≥ 3. Both results are based on establishing the number of layers in near optimal permutations using a layer-merging technique.