Josefran de Oliveira Bastos ; Leonardo Nagami Coregliano
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Packing densities of layered permutations and the minimum number of
monotone sequences in layered permutations
Packing densities of layered permutations and the minimum number of
monotone sequences in layered permutations
Authors: Josefran de Oliveira Bastos ; Leonardo Nagami Coregliano
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Josefran de Oliveira Bastos;Leonardo Nagami Coregliano
In this paper, we present two new results of layered permutation densities.
The first one generalizes theorems from Hästö (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).