Bastos, Josefran de Oliveira and Coregliano, Leonardo Nagami - Packing densities of layered permutations and the minimum number of monotone sequences in layered permutations

dmtcs:1313 - Discrete Mathematics & Theoretical Computer Science, June 23, 2016, Vol. 18 no. 2, Permutation Patterns 2015
Packing densities of layered permutations and the minimum number of monotone sequences in layered permutations

Authors: Bastos, Josefran de Oliveira and Coregliano, Leonardo Nagami

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).


Source : oai:arXiv.org:1510.07312
Volume: Vol. 18 no. 2, Permutation Patterns 2015
Section: Permutation Patterns
Published on: June 23, 2016
Submitted on: June 22, 2016
Keywords: Mathematics - Combinatorics,Computer Science - Discrete Mathematics,05A05 (Primary), 05D99 (Secondary)


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