Efficient counting of permutation patterns via double posets

2024-08-15

AI Summary

This work addresses the efficient counting of permutation patterns. We introduce a unified modeling and algorithmic framework based on double posets. Our key innovation is the formalization of corner trees as a special class of double posets—termed *twin-tree double posets*—and their generalization to a broader family of tree-structured double posets. Building on this structural characterization, we extend the Even–Zohar–Leng algorithm to this new setting, yielding a subquadratic counting algorithm with time complexity $O(n^{5/3})$. This approach substantially transcends the expressive limitations of the original corner tree model, significantly expanding the class of permutation patterns amenable to efficient enumeration. The framework provides a novel structural tool and algorithmic paradigm for permutation pattern theory and combinatorial enumeration.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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}).

Problem

• Efficient counting of permutation patterns using double posets
• Generalizing corner tree counting via twin-tree double posets
• Extending algorithm for faster permutation pattern counting

Innovation

• Utilizes twin-tree double posets for pattern counting
• Generalizes algorithm to tree double posets
• Enables O(n^{5/3}) permutation counting