Given permutations τ and π, the permutation pattern (PP) problem is to decide whether π occurs in τ as an order-isomorphic subsequence. Although an FPT algorithm is known for PP parameterized by the size of the pattern |π| [Guillemot and Marx 2014], the high complexity of this algorithm makes it impractical for most instances. In this paper we approach the PP problem from k-track permutations, i.e. those permutations that are the union of k increasing patterns or, equivalently, those permutation that avoid the decreasing pattern (k+1)k…1. Recently, k-track permutations have been shown to be central combinatorial objects in the study of the PP problem. Indeed, the PP problem is NP-complete when π is 321-avoiding and τ is 4321-avoiding but is solvable in polynomial-time if both π and τ avoid 321. We propose and implement an exact algorithm, FPT for parameters k and |π|, which allows to solve efficiently some large instances.