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Completing the enumeration of inversion sequences avoiding triples of relations

Nathan Britt, Nicholas Beaton

TL;DR

The paper completes the enumeration of 14 previously unresolved inversion-sequence classes avoiding triples of binary relations, by constructing generating trees that grow either on the left or on the right and tracking key statistics such as the maximum and premaximum. For many classes, the authors derive systems of functional equations and, via the kernel method, obtain algebraic generating functions; several classes remain non-algebraic, with detailed asymptotics provided where possible. The work also establishes left/right-growth strategies, highlights Wilf-equivalences, and offers a rich set of open questions for extending pattern-avoidance analyses to larger families. Overall, it advances the understanding of pattern-avoidance in inversion sequences and demonstrates powerful generating-tree techniques coupled with the kernel method.

Abstract

An inversion sequence of length $n$ is an integer sequence $(a_1, \ldots, a_n)$ such that $0 \le a_i < i$ for all $i$. The study of pattern-avoiding inversion sequences was initiated in 2015 by Mansour and Shattuck and in 2016 by Corteel, Martinez, Savage and Weselcouch. Martinez and Savage later defined a new type of pattern, a triple of binary relations, of which there are currently 14 uncounted avoidance classes. We complete the enumeration for all of these classes using generating tree methods "growing on the left" and "growing on the right". For many of these classes we are able to find algebraic generating functions. We also discuss the asymptotic behaviour of the counting sequences.

Completing the enumeration of inversion sequences avoiding triples of relations

TL;DR

The paper completes the enumeration of 14 previously unresolved inversion-sequence classes avoiding triples of binary relations, by constructing generating trees that grow either on the left or on the right and tracking key statistics such as the maximum and premaximum. For many classes, the authors derive systems of functional equations and, via the kernel method, obtain algebraic generating functions; several classes remain non-algebraic, with detailed asymptotics provided where possible. The work also establishes left/right-growth strategies, highlights Wilf-equivalences, and offers a rich set of open questions for extending pattern-avoidance analyses to larger families. Overall, it advances the understanding of pattern-avoidance in inversion sequences and demonstrates powerful generating-tree techniques coupled with the kernel method.

Abstract

An inversion sequence of length is an integer sequence such that for all . The study of pattern-avoiding inversion sequences was initiated in 2015 by Mansour and Shattuck and in 2016 by Corteel, Martinez, Savage and Weselcouch. Martinez and Savage later defined a new type of pattern, a triple of binary relations, of which there are currently 14 uncounted avoidance classes. We complete the enumeration for all of these classes using generating tree methods "growing on the left" and "growing on the right". For many of these classes we are able to find algebraic generating functions. We also discuss the asymptotic behaviour of the counting sequences.
Paper Structure (21 sections, 20 theorems, 65 equations, 3 figures, 1 table)

This paper contains 21 sections, 20 theorems, 65 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Inversion sequences grown on the right
  3. Class 1176: $(>,\leq,\neq) \equiv (100, 102, 201)$
  4. Class 1253: $(>, \neq, \neq) \equiv (102, 201, 210)$
  5. Class 1016: $(>,-,\neq) \equiv (100, 102, 201, 210)$
  6. Class 830: $(\ne, >, \ge) \equiv (010, 120, 210)$
  7. Class 2106: $(>, \le, \ge) \equiv (100, 101, 201)$
  8. Inversion sequences grown on the left
  9. Class 663A: $(-,\neq,\geq) \equiv (010,101,110,120,201,210)$
  10. Class 1420: $(-,-,>) \equiv (100,110,120,201,210)$
  11. Class 1833A: $(-,\neq,>) \equiv (110,120,201,210)$
  12. Class 733: $(\neq,\neq,\geq) \equiv (010,101,120,201,210)$
  13. Class 214: $(-,\geq,\geq) \equiv (000,010,100,110,120,210)$
  14. Class 1509: $(-,\geq,>) \equiv (100,110,120,210)$
  15. Class 1953A: $(-,>,>) \equiv (110,120,210)$
  16. ...and 6 more sections

Key Result

Lemma 1

An inversion sequence $a \in \mathcal{I}$ avoids the patterns 102 and 201 if and only if it can be factored as $a = p_1 \circ p_2 \circ p_3$ where:

Figures (3)

  • Figure 1: The progression of a (100, 102, 201)-avoiding inversion sequence growing on the right, as it graduates through the sets $\mathcal{B}$, then $\mathcal{C}$, then $\mathcal{D}$, and landing finally in $\mathcal{E}^{(2)}$.
  • Figure 2: A progression of inversion sequences described in the rule $\Omega_{(\ne, -, \ge)}$. Each positive term is raised, some zeros may be raised, and a zero is prepended. In this case, only the sequence containing blue nodes avoids $(\ne, -, \ge)$.
  • Figure 3: An illustration of the growth of Class 759 using commitments. The sequence in (a) is in $\mathcal{A}$, with $p=7$. In (b) we have incremented a 0 not on the left of $P$, to get something in $\mathcal{B}$ with $k=6$. The number of commitments is $c=4$, with the word on the alphabet $\{1,2,3,4\}$ avoiding 212, 112 and 213. In (c) we have fulfilled the first commitment.

Theorems & Definitions (27)

  • Lemma 1: testart_completing_2024
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Proposition 4
  • Proposition 5
  • ...and 17 more