Permutation Reconstruction from Minors

We consider the problem of permutation reconstruction, which is 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 M-k(p) with an underlying set M-k(l)(p). We study the question of when we can reconstruct p from its multiset or its set of minors. We prove there exists an N-k for every k such that any permutation with length at least N-k is reconstructible from its multiset of (n-k)-minors. We find the bounds N-k > k+log(2) k and N-k < (k2)/(4) + 2k + 4. For the number N-k(l), giving the minimal length for permutations to be reconstructible for their sets of (n-k)-minors, we have the bound N-k(l) > 2k. We work towards analogous bounds in the case when we restrict ourselves to layered permutations.