Efficient counting of permutation patterns via double posets

TL;DR

This paper generalizes corner trees, a tool for counting permutation patterns, by introducing twin-tree double posets and extending algorithms to a broader class of posets, enabling efficient counting.

Contribution

It identifies twin-tree double posets as a new class that generalizes corner trees and extends counting algorithms to these structures.

Findings

Enlarged class of posets for permutation counting

Extended algorithm for tree double posets

Maintained efficient counting complexity of O(n^{5/3})

Abstract

Corner trees, introduced in "Even-Zohar and Leng, 2021, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms", allow for the efficient counting of certain permutation patterns. Here we identify corner trees as a subset of finite (strict) double posets, which we term twin-tree double posets. They are contained in both twin double posets and tree double posets, giving candidate sets for generalizations of corner tree countings. We provide the generalization of an algorithm proposed by Even-Zohar/Leng to a class of tree double posets, thereby enlarging the space of permutations that can be counted in O(n^{5/3}).