Counting Permutation Patterns with Multidimensional Trees

We consider the well-studied pattern-counting problem: given a permutation π ∈ Sn and an integer k > 1, count the number of order-isomorphic occurrences of every pattern τ ∈ S_k in π. Our first result is an Õ(n²)-time algorithm for k = 6 and k = 7. The proof relies heavily on a new family of graphs that we introduce, called pattern-trees. Every such tree corresponds to an integer linear combination of permutations in S_k, and is associated with linear extensions of partially ordered sets. We design an evaluation algorithm for these combinations, and apply it to a family of linearly-independent trees. For k = 8, we show a barrier: the subspace spanned by trees in the previous family has dimension exactly |S8|− 1, one less than required. Our second result is an Õ(n^7/4)-time algorithm for k = 5. This algorithm extends the framework of pattern-trees by speeding-up their evaluation in certain cases. A key component of the proof is the introduction of pair-rectangle-trees, a data structure for dominance counting.