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On minimal pattern-containing inversion sequences

Benjamin Testart

TL;DR

The paper introduces minimal pattern-containing inversion sequences for a pattern $\rho$ and develops a rigorous framework around $\mathcal{I}_n$ and $\mathcal{P}_n$, including the statistics $mdd(\rho)$, $dist$, and $Sat$, to characterize $\rho$-minimal sequences. It proves a structural characterization via filler-entry conditions around each occurrence of $\rho$, yielding sharp length bounds $|\rho|+mdd(\rho) \le |\sigma| \le |\rho|+2\,mdd(\rho)$ and the aggregate bound $|\sigma| \le 3|\rho| - 2$, with explicit constructions showing tightness. The authors provide enumerative results by bijecting $\rho$-minimal inversion sequences to colored-prefix trees for patterns with $\rho_1 = mdd(\rho)$, and compute counts for patterns up to length $5$, alongside a broader enumeration of $\mathcal{I} \cap \mathcal{P}$ via non-plane increasing trees, linking to poly-Bernoulli numbers of type $C$. They also introduce practical exhaustive-generation methods and a new data descriptor, ISBT, to classify minimal sequences, culminating in open questions about Wilf-equivalence, longest minimal sequences, and deeper connections to known poly-Bernoulli frameworks. These results advance pattern-avoidance in inversion sequences and connect combinatorial structures with classic integer sequences.

Abstract

We introduce the notion of minimal inversion sequences for a pattern $ρ$, which form the smallest set of inversion sequences whose avoidance is equivalent to the avoidance of $ρ$ for inversion sequences. We give a characterization of $ρ$-minimal inversion sequences based on the occurrences of the pattern $ρ$ they contain, and use it to find upper and lower bounds on the lengths of $ρ$-minimal inversion sequences. We provide some enumerative results on the exact number of minimal inversion sequences for some patterns, through a bijection with increasing trees, and some exhaustive generation. Lastly, we enumerate inversion sequences which are equal to their reduction, and find an interesting connection with poly-Bernoulli numbers.

On minimal pattern-containing inversion sequences

TL;DR

The paper introduces minimal pattern-containing inversion sequences for a pattern and develops a rigorous framework around and , including the statistics , , and , to characterize -minimal sequences. It proves a structural characterization via filler-entry conditions around each occurrence of , yielding sharp length bounds and the aggregate bound , with explicit constructions showing tightness. The authors provide enumerative results by bijecting -minimal inversion sequences to colored-prefix trees for patterns with , and compute counts for patterns up to length , alongside a broader enumeration of via non-plane increasing trees, linking to poly-Bernoulli numbers of type . They also introduce practical exhaustive-generation methods and a new data descriptor, ISBT, to classify minimal sequences, culminating in open questions about Wilf-equivalence, longest minimal sequences, and deeper connections to known poly-Bernoulli frameworks. These results advance pattern-avoidance in inversion sequences and connect combinatorial structures with classic integer sequences.

Abstract

We introduce the notion of minimal inversion sequences for a pattern , which form the smallest set of inversion sequences whose avoidance is equivalent to the avoidance of for inversion sequences. We give a characterization of -minimal inversion sequences based on the occurrences of the pattern they contain, and use it to find upper and lower bounds on the lengths of -minimal inversion sequences. We provide some enumerative results on the exact number of minimal inversion sequences for some patterns, through a bijection with increasing trees, and some exhaustive generation. Lastly, we enumerate inversion sequences which are equal to their reduction, and find an interesting connection with poly-Bernoulli numbers.
Paper Structure (11 sections, 10 theorems, 20 equations, 3 figures, 3 tables)

This paper contains 11 sections, 10 theorems, 20 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Motivation and outline
  4. Definitions, notation, and simple observations
  5. Structure of minimal inversion sequences
  6. Optimality
  7. Enumeration of minimal inversion sequences
  8. Patterns beginning by a large value
  9. Exhaustive generation for small patterns
  10. Enumeration of I ∩ P
  11. Open questions

Key Result

Proposition 4

Let $n,k \geq 1$, let $\rho \in \mathcal{P}_k$, and let $\sigma \in \mathcal{I}_n \cap \mathcal{P}_n$ be a sequence containing the pattern $\rho$. Then $\sigma$ is $\rho$-minimal if and only if for any set $R \subseteq [1,n]$ such that $\rho' := (\sigma_{i})_{i \in R}$ is an occurrence of $\rho$, bo

Figures (3)

  • Figure 1: On the left, the sequence 20260 of maximum diagonal difference 3. On the right, its reduction 10120 of maximum diagonal difference 1.
  • Figure 2: The coloured inversion sequence $\alpha = \textcolor{red}{\overline{0}} \textcolor{red}{\overline{0}} \textcolor{blue}{\underline{2}} \textcolor{red}{\overline{3}} \textcolor{red}{\overline{0}} \textcolor{red}{\overline{3}}$ and the Cayley permutation $\rho = 4540312$ together form the $\rho$-minimal inversion sequence ${\sigma = 0023036761524}$.
  • Figure 3: Illustration of the function $\phi$.

Theorems & Definitions (23)

  • Definition 1: From Kotsireas_Mansour_Yildirim_2024
  • Claim 2: From Kotsireas_Mansour_Yildirim_2024
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Corollary 7
  • ...and 13 more