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Enumerations of some pattern-avoiding Fishburn permutations

Yujie Du, Philip B. Zhang

Abstract

In this paper, we prove two conjectures of Egge on the enumeration of several classes of pattern-avoiding Fishburn permutations. Our results include enumerating Fishburn permutations avoiding pattern 321 and one of the following three types of classical patterns: a pattern of size 4, two patterns of size 4, or a pattern of size 5.

Enumerations of some pattern-avoiding Fishburn permutations

Abstract

In this paper, we prove two conjectures of Egge on the enumeration of several classes of pattern-avoiding Fishburn permutations. Our results include enumerating Fishburn permutations avoiding pattern 321 and one of the following three types of classical patterns: a pattern of size 4, two patterns of size 4, or a pattern of size 5.
Paper Structure (19 sections, 35 theorems, 71 equations, 1 figure, 1 table)

This paper contains 19 sections, 35 theorems, 71 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Avoiding 321 and a classical pattern of size 4
  3. Enumerating $F_{n}(321,1243)$
  4. Enumerating $F_{n}(321,2134)$
  5. Enumerating $F_{n}(321,1324)$
  6. Avoiding 321 and two classical patterns of size 4
  7. Enumerating $F_{n}(321,1423,2143)$
  8. Enumerating $F_{n}(321,3142,2143)$
  9. Enumerating $F_{n}(321,2143,3124)$
  10. Enumerating $F_{n}(321,2143,4123)$
  11. Enumerating $F_{n}(321,1423,3124)$
  12. Enumerating $F_{n}(321,1423,4123)$
  13. Enumerating $F_{n}(321,3124,4123)$
  14. Avoiding 321 and a classical pattern of size 5
  15. Enumerating $F_n(321,14253)$
  16. ...and 4 more sections

Key Result

Lemma 1.1

Let $\pi\in F_n(321)$. Then either $\pi_1=1$ or $\pi_2=1$.

Figures (1)

  • Figure :

Theorems & Definitions (71)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • proof : Proof of Theorem \ref{['1243']}
  • ...and 61 more