Packing densities of layered permutations and the minimum number of monotone sequences in layered permutations

TL;DR

This paper advances understanding of layered permutation densities by generalizing packing theorems and establishing asymptotic minimum densities of monotone sequences, connecting to longstanding conjectures.

Contribution

It generalizes permutation packing results to broader layered structures and determines asymptotic minimum densities of monotone sequences in layered permutations.

Findings

Permutation packing densities are computed for specific layered structures.

Minimum density of monotone sequences of length k+1 is asymptotically 1/k^k.

Results support conjectures relating to non-layered permutation problems.

Abstract

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).