  In this paper, we present two new results of layered permutation densities.
The first one generalizes theorems from H\"{a}st\"{o} (2003) and Warren (2004)
to compute the permutation packing of permutations whose layer sequence
is~$(1^a,\ell_1,\ell_2,\ldots,\ell_k)$ with~$2^a-a-1\geq k$ (and similar
permutations). As a second result, we prove that the minimum density of
monotone sequences of length~$k+1$ in an arbitrarily large layered permutation
is asymptotically~$1/k^k$. This value is compatible with a conjecture from
Myers (2003) for the problem without the layered restriction (the same problem
where the monotone sequences have different lengths is also studied).
