Consecutive and quasi-consecutive patterns: $mathrm{des}$-Wilf classifications and generating functions

2025-02-14

AI Summary

This paper investigates the distribution of descents (des) in permutations avoiding quasi-consecutive patterns, completing the des-Wilf equivalence classification for all quasi-consecutive patterns of length at most four. Methodologically, it constructs several descent-preserving bijections—thereby filling two previously incomplete bijective arguments in the literature—and integrates descent statistics, a generalized run theorem, and generating function derivations. The main contributions are: (i) the first comprehensive des-Wilf classification of all 127 quasi-consecutive patterns of length ≤ 4, partitioned into 23 equivalence classes; and (ii) exact bivariate generating functions for the two singleton classes. This work significantly advances the Wilf classification theory for consecutive and quasi-consecutive patterns by unifying combinatorial bijections with analytic generating function techniques.

Technology Category

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

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as established in this paper, we obtain a complete $des$-Wilf classification for quasi-consecutive patterns of length up to 4. For equivalence classes containing more than one pattern, we construct various descent-preserving bijections to establish the equivalences, which lead to the provision of proper versions of two incomplete bijective arguments previously published in the literature. Additionally, for two singleton classes, we derive explicit bivariate generating functions using the generalized run theorem.

Problem

• Classify quasi-consecutive patterns up to length 4.
• Establish descent-preserving bijections for equivalence classes.
• Derive bivariate generating functions for singleton classes.

Innovation

• des-Wilf classification
• descent-preserving bijections
• bivariate generating functions