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 M ′ k(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 < k 2 4 +2k+4. For the number N ′ k, giving the minimal length for permutations to be reconstructible from their sets of (n − k)-minors, we have the bound N ′ k > 2k. We work towards analogous bounds in the case when we restrict ourselves to layered permutations.