Permutations resilient to deletions

Let σ be a permutation on [n] = {1, 2, . . . , n} which can be written in two-line notation, and let φ : [n] → S be a bijection. Construct τ (resp. β) by replacing the elements in σ as dictated by φ and then deleting up to d elements in the top (resp. bottom) line and contracting the result, making sure no symbol is deleted in both lines. The permutation σ is d-resilient if τ and β always uniquely determine φ (or equivalently, determine where the deletions in the top and bottom lines occurred). Necessary and sufficient conditions for a permutation to be d-resilient are established in terms of whether a family of auxiliary graphs are acyclic. Also, constructions are given for d-resilient permutations which have size n exponential in d, this is best possible. It is further shown that for every fixed d and sufficiently large n a positive portion of all permutations of n elements are d-resilient.