Counting Permutation Patterns with Multidimensional Trees

TL;DR

This paper introduces new algorithms and data structures, such as pattern-trees and pair-rectangle-trees, to efficiently count permutation patterns for specific pattern sizes, advancing the computational understanding of pattern counting.

Contribution

The paper presents novel algorithms for counting permutation patterns of sizes 5, 6, and 7, utilizing new graph structures called pattern-trees and pair-rectangle-trees, and identifies a complexity barrier for size 8.

Findings

Achieved (n^2) time for k=6,7

Developed an (n^{7/4}) time algorithm for k=5

Identified a dimension barrier at k=8 for the pattern-tree approach.

Abstract

We consider the well-studied pattern counting problem: given a permutation $\pi \in \mathbb{S}_n$ and an integer $k > 1$, count the number of order-isomorphic occurrences of every pattern $\tau \in \mathbb{S}_k$ in $\pi$. Our first result is an $\widetilde{\mathcal{O}}(n^2)$-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 $\mathbb{S}_k$, 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 $|\mathbb{S}_8| - 1$, one less than required. Our second result is an $\widetilde{\mathcal{O}}(n^{7/4})$-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.