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

Yan Wang, Qi Fang, Shishuo Fu, Sergey Kitaev, Haijun Li

TL;DR

The paper advances the study of permutation pattern avoidance by focusing on the descent statistic for quasi-consecutive patterns. It introduces the Structure Theorem, which ties the generating function for a quasi-consecutive pattern $p=\sigma k$ to the generating function for the base consecutive pattern $\sigma$, enabling a unified treatment of descent distributions. Through a combination of descent-preserving bijections, the generalized run theorem, and partial differential equations, the authors achieve a complete des-Wilf classification for patterns of length up to four and provide explicit generating functions for several singleton classes. The work deepens connections between pattern avoidance, enumerative combinatorics, and classical structures such as set partitions and Narayana numbers, and lays groundwork for future explorations of other statistics and longer patterns.

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.

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

TL;DR

The paper advances the study of permutation pattern avoidance by focusing on the descent statistic for quasi-consecutive patterns. It introduces the Structure Theorem, which ties the generating function for a quasi-consecutive pattern to the generating function for the base consecutive pattern , enabling a unified treatment of descent distributions. Through a combination of descent-preserving bijections, the generalized run theorem, and partial differential equations, the authors achieve a complete des-Wilf classification for patterns of length up to four and provide explicit generating functions for several singleton classes. The work deepens connections between pattern avoidance, enumerative combinatorics, and classical structures such as set partitions and Narayana numbers, and lays groundwork for future explorations of other statistics and longer patterns.

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 -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.

Paper Structure

This paper contains 10 sections, 20 theorems, 49 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Structure Theorem and classification of length-3 quasi-consecutive patterns
  3. Proof of Theorem \ref{['structure thm']}
  4. Classifications for lengths 2 and 3
  5. des-Wilf classification of length 4 quasi-consecutive patterns
  6. Generating functions
  7. The run theorem
  8. The generating functions involving partial differential equations
  9. Conclusion
  10. appendix

Key Result

Theorem 1.1

Suppose $\sigma\sim\sigma'$ are two Wilf-equivalent consecutive patterns of length $k-1$, then

Figures (4)

  • Figure 1: A decomposition of $\pi \in \mathfrak{S}_{n+1}(p)$
  • Figure 2: An illustration of choosing $\pi_{j_k}$ and $\pi_{i_k}$, $1\le k\le l$
  • Figure 3: Run network for permutations avoiding consecutive pattern $\underline{123}$.
  • Figure 4: Position of $1$ in $\pi \in \mathfrak{S}_{n+1}(\underline{132})$

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2: Structure Theorem
  • proof : Proof of Theorem \ref{['structure thm']}
  • Corollary 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 28 more