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Pattern avoidance in revised ascent sequences

Robin D. P. Zhou

TL;DR

This work introduces revised ascent sequences as Cayley permutations with a leftmost-occurrence criterion tied to ascent bottoms, and proves a bijection with ordinary ascent sequences via a hat-map-inspired construction, establishing that revised ascent sequences are counted by Fishburn numbers. It then systematically analyzes the pattern-avoidance landscape for single-pattern restrictions, deriving explicit counts, structural descriptions, and generating-functions for a broad array of patterns through bijections, generating trees, and the kernel method. The results reveal connections to Catalan and Bell numbers, Stirling numbers, and Dyck-path families, and highlight rich Wilf-equivalences and structural decompositions within the revised ascent framework. The work also outlines several open problems and future directions, including the pattern 111, direct bijections to other Fishburn structures, and the exploration of longer-pattern avoidance. Overall, the paper expands the combinatorial landscape surrounding Fishburn-type objects by introducing revised ascent sequences and detailing their single-pattern avoidance behavior with diverse enumerative tools.

Abstract

Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only if it serves as an ascent bottom. We construct a bijection between ascent sequences and revised ascent sequences by adapting the classic hat map, which transforms ascent sequences into modified ascent sequences. Additionally, we investigate revised ascent sequences that avoid a single pattern, leading to a wealth of enumerative results. Our main techniques include the use of bijections, generating trees, generating functions, and the kernel method.

Pattern avoidance in revised ascent sequences

TL;DR

This work introduces revised ascent sequences as Cayley permutations with a leftmost-occurrence criterion tied to ascent bottoms, and proves a bijection with ordinary ascent sequences via a hat-map-inspired construction, establishing that revised ascent sequences are counted by Fishburn numbers. It then systematically analyzes the pattern-avoidance landscape for single-pattern restrictions, deriving explicit counts, structural descriptions, and generating-functions for a broad array of patterns through bijections, generating trees, and the kernel method. The results reveal connections to Catalan and Bell numbers, Stirling numbers, and Dyck-path families, and highlight rich Wilf-equivalences and structural decompositions within the revised ascent framework. The work also outlines several open problems and future directions, including the pattern 111, direct bijections to other Fishburn structures, and the exploration of longer-pattern avoidance. Overall, the paper expands the combinatorial landscape surrounding Fishburn-type objects by introducing revised ascent sequences and detailing their single-pattern avoidance behavior with diverse enumerative tools.

Abstract

Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only if it serves as an ascent bottom. We construct a bijection between ascent sequences and revised ascent sequences by adapting the classic hat map, which transforms ascent sequences into modified ascent sequences. Additionally, we investigate revised ascent sequences that avoid a single pattern, leading to a wealth of enumerative results. Our main techniques include the use of bijections, generating trees, generating functions, and the kernel method.
Paper Structure (14 sections, 25 theorems, 57 equations, 1 table)

This paper contains 14 sections, 25 theorems, 57 equations, 1 table.

Table of Contents

  1. Introduction
  2. A bijection between $\mathcal{A}_n$ and $\hat{\mathcal{B}}_{n+1}$
  3. Pattern avoidance in $\hat{\mathcal{B}}_n$
  4. Patterns of length $2$ and pattern $212$
  5. Pattern $221$
  6. Pattern $312$
  7. Pattern $122$
  8. Patterns $321$ and $231$
  9. Pattern $213$
  10. Patterns $211$ and $121$
  11. Pattern $112$
  12. Pattern $123$
  13. Pattern $132$
  14. Final remarks and future directions

Key Result

Lemma 2.1

Let $n\geq 2$ and let $x = x_1x_2\cdots x_n$ be an ascent sequence. Then we have $\mathrm{asctop}(x) = \max(\eta(x))$ and $\eta(x) = \mathrm{Add}(\eta(x'),x_n)$, where $x' = x_1x_2\cdots x_{n-1}$.

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Proposition 3.6
  • Proposition 3.7
  • ...and 15 more