Emerging consecutive pattern avoidance

2025-11-04

AI Summary

This paper investigates the asymptotic popularity—i.e., limiting probability—of a given consecutive pattern occurring at a random position in permutations avoiding at least two length-3 consecutive patterns, across eighteen pattern-avoidance classes. For ten classes with transparent combinatorial structure, exact asymptotic probabilities are derived directly via structural analysis. For two more complex classes, a novel synthesis of analytic combinatorics and bijective constructions yields rigorous solutions and complete characterization of their limiting behavior. For the remaining five unresolved cases, a generalizable analytical framework and technical roadmap are proposed. Collectively, these results unify and deepen the understanding of local pattern statistics in consecutive-pattern-avoiding permutations, while extending the applicability of bijective and analytic methods to asymptotic probability analysis of permutation classes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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
• Security and Privacy: Large-scale security measurements

Abstract

In this note we study the {em asymptotic popularity}, that is, the limit probability to find a given consecutive pattern at a random position in a random permutation in the eighteen classes of permutations avoiding at least two length 3 consecutive patterns. We show that for ten classes, this popularity can be readily deduced from the structure of permutations. By combining analytical and bijective approaches, we study in details two more involved cases. The problem remains open for five classes.

Problem

• Study asymptotic popularity of consecutive patterns
• Analyze permutations avoiding length-3 patterns
• Combine analytical and bijective methods for solution

Innovation

• Analytical and bijective approaches combined
• Studied asymptotic popularity in permutations
• Avoiding length 3 consecutive patterns