Determining Lower Bounds for Packing Densities of Non-layered Patterns Using Weighted Templates

Abstract

The packing density of a permutation pattern $\pi$ is the limiting value, ${n}$ $\rightarrow$ $\infty$, of the maximum proportion of subsequences of $\sigma$ $\in$ ${S_{n}}$ that are order-isomorphic to $\pi$. We generalize methods for obtaining lower bounds for the packing density of any pattern and demonstrate the methods' usefulness when patterns are non-layered.