Permutation Reconstruction from Minors

We consider the problem of permutation reconstruction, which is a variant of graph reconstruction. Given a permutation p of length n, we delete k of its entries in each possible way to obtain n k subsequences. We renumber the sequences from 1 to n-k preserving the relative size of the elements to form (n-k)-minors. These minors form a multiset Mk(p) with an underlying set Mk(p). We study the question of when we can reconstruct p from its multiset or its set of minors. We prove there exists an Nk for every k such that any permutation with length at least Nk is reconstructible from its multiset of (n-k)-minors. We find the bounds Nk > k+log2 k and Nk < k2 4 +2k+4. For the number Nk, giving the minimal length for permutations to be reconstructible from their sets of (n - k)-minors, we have the bound Nk > 2k. We work towards analogous bounds in the case when we restrict ourselves to layered permutations.

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: Mariana Raykova
• Keywords: COMBINATORICS 2006, Graph Reconstruction, Permutation Reconstruction, Permutations