On uniquely k-determined permutations

There are several approaches to study occurrences of consecutive patterns in permutations such as the inclusion-exclusion method, the tree representations of permutations, the spectral approach and others. We propose yet another approach to study occurrences of consecutive patterns in permutations. The approach is based on considering the graph of patterns overlaps, which is a certain subgraph of the de Bruijn graph. While applying our approach, the notion of a uniquely $k$-determined permutation appears. 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 applying the transfer matrix method possible for determining the number of uniquely $k$-determined permutations.