On uniquely k-determined permutations

Motivated by a new point of view to study occurrences of consecutive patterns in permutations, we introduce the notion of uniquely k-determined permutations. We give two criteria for a permutation to be uniquely k-determined: one in terms of the distance between two consecutive elements in a permutation, and the other one in terms of directed hamiltonian paths in the certain graphs called path-schemes. Moreover, we describe a finite set of prohibitions that gives the set of uniquely k-determined permutations. Those prohibitions make the application of the transfer matrix method possible for determining the number of uniquely k-determined permutations.