Improvements on Permutation Reconstruction from Minors

TL;DR

This paper investigates the permutation reconstruction problem from minors, establishing new bounds on the minimal minor size needed for reconstruction, which improves understanding of permutation parameters and their limits.

Contribution

The authors derive new asymptotic bounds for the minimal minor size required to reconstruct permutations, advancing theoretical understanding of permutation reconstruction.

Findings

Bounds on the minimal minor size are between exponential and square-root functions.

Improved bounds on the parameter $N_d$ for permutation reconstruction.

The results refine previous logarithmic and quadratic bounds.

Abstract

We study the reconstruction problem of permutation sequences from their $k$-minors, which are subsequences of length $k$ with entries renumbered by $1,2,\ldots,k$ preserving order. We prove that the minimum number $k$ such that any permutation of length $n$ can be reconstructed from the multiset of its $k$-minors is between $\exp{(\Omega(\sqrt{\ln n}))}$ and $O(\sqrt{n\ln n})$. These results imply better bounds of a well-studied parameter $N_d$, which is the smallest number such that any permutation of length $n\ge N_d$ can be reconstructed by its $(n-d)$-minors. The new bounds are $ d+\exp(\Omega(\sqrt{\ln d}))<N_d<d+O(\sqrt{d\ln d})$ asymptotically, and the previous bounds were $d+\log_2 d<N_d<d^2/4+2d+4$.