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Completing the enumeration of inversion sequences avoiding one or two patterns of length 3

Benjamin Testart

TL;DR

This work completes the enumeration of inversion sequences avoiding one or two patterns of length 3 by introducing four constructive approaches, including new shifted-inversion-sequence techniques, to resolve 24 classes. Central to the methodology are generating trees grown on the right, the ‘split around the maximum’ and ‘split around the minimum’ decompositions, and pattern-avoidance reductions to constrained words, all of which yield recurrences and, in several cases, algebraic generating functions. The authors prove several conjectures (notably via kernel-method analyses) and provide explicit generating functions for key pairs such as $\mathcal{I}(102,201)$ and $\mathcal{I}(102,210)$, while also discussing asymptotic growth rates and broader implications for pattern-avoidance in Lehmer-code-like sequences. The results establish a coherent framework for pattern-avoiding inversion sequences and open avenues for understanding algebraicity and growth rates in related combinatorial classes.

Abstract

We present four constructions of inversion sequences, and use them to compute the enumeration sequences of 24 classes of pattern-avoiding inversion sequences. This completes the enumeration of inversion sequences avoiding one or two patterns of length 3. Some of our constructions are based on generating trees. Others involve pattern-avoiding words, which we also count using generating trees. To solve some of these cases, we introduce a generalization of inversion sequences, which we call shifted inversion sequences. Lastly, we briefly discuss the asymptotics of pattern-avoiding inversion sequences, focusing on their exponential or super-exponential behavior.

Completing the enumeration of inversion sequences avoiding one or two patterns of length 3

TL;DR

This work completes the enumeration of inversion sequences avoiding one or two patterns of length 3 by introducing four constructive approaches, including new shifted-inversion-sequence techniques, to resolve 24 classes. Central to the methodology are generating trees grown on the right, the ‘split around the maximum’ and ‘split around the minimum’ decompositions, and pattern-avoidance reductions to constrained words, all of which yield recurrences and, in several cases, algebraic generating functions. The authors prove several conjectures (notably via kernel-method analyses) and provide explicit generating functions for key pairs such as and , while also discussing asymptotic growth rates and broader implications for pattern-avoidance in Lehmer-code-like sequences. The results establish a coherent framework for pattern-avoiding inversion sequences and open avenues for understanding algebraicity and growth rates in related combinatorial classes.

Abstract

We present four constructions of inversion sequences, and use them to compute the enumeration sequences of 24 classes of pattern-avoiding inversion sequences. This completes the enumeration of inversion sequences avoiding one or two patterns of length 3. Some of our constructions are based on generating trees. Others involve pattern-avoiding words, which we also count using generating trees. To solve some of these cases, we introduce a generalization of inversion sequences, which we call shifted inversion sequences. Lastly, we briefly discuss the asymptotics of pattern-avoiding inversion sequences, focusing on their exponential or super-exponential behavior.
Paper Structure (41 sections, 60 theorems, 161 equations, 5 figures, 3 tables)

This paper contains 41 sections, 60 theorems, 161 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Context and summary of results
  4. Notation and preliminaries
  5. Generating trees growing on the right
  6. Method
  7. The pair {000, 100}
  8. The pair {102, 201}
  9. Removing the maximum
  10. Method
  11. The pair {000, 102}
  12. The pair {102, 210}
  13. The pairs {000, 201} or {000, 210}
  14. The pair {100, 110}
  15. The pair {100, 101}
  16. ...and 26 more sections

Key Result

Proposition 1

For any set of patterns $P$, for all $n,k \in \mathbb N$,

Figures (5)

  • Figure 1: First five levels of the combinatorial generating tree growing on the right for $\mathcal{I}(10)$.
  • Figure 2: First five levels of the generating tree described by the succession rule $\Omega_{\text{Cat}}$.
  • Figure 3: Part of the generating tree described by the succession rule $\Omega_{\{102,201\}}$. Phantom objects are colored in gray.
  • Figure 4: First five levels of the generating tree described by the succession rule $\Omega_{\{000,102\}}$.
  • Figure 5: Part of the generating tree described by the succession rule $\Omega_{\{102,210\}}$.

Theorems & Definitions (134)

  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • proof
  • Corollary 4
  • Corollary 5
  • Lemma 6
  • proof
  • ...and 124 more