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On two conjectures about pattern avoidance of cyclic permutations

Junyao Pan

TL;DR

The paper studies pattern avoidance for cyclic permutations in both one-line and standard cycle forms, focusing on the enumerative function $a_n(\sigma_1,...,\sigma_k; \tau)$. It resolves two Archer et al. conjectures by constructing bijections and structural analyses: (i) for patterns $\tau=\tau_1\cdots\tau_{k-2}21$ with $k\ge4$, $a_n(\tau,4321;213)=F_{2n-3}$, and (ii) $a_n(1324,1423;213)=\binom{n}{3}+1$. The authors also derive several symmetric and additional enumerative results, including $a_n(3421,4321;213)=a_n(4312,4321;312)=F_{2n-3}$, $a_n(4321;213)=F_{2n-3}$, $a_n(4312,4321;213)=2^{n-2}$, and $a_n(3412,4321;213)=P_{n-1}$, as well as a detailed proof converting $a_n(1324,1423;213)$ to $\binom{n}{3}+1$ via decompositions and size-preserving bijections. These results enhance understanding of pattern avoidance in cyclic permutations and connect to classical sequences such as Fibonacci and Pell numbers.

Abstract

Let $π$ be a cyclic permutation that can be expressed in its one-line form as $π= π_1π_2 \cdot\cdot\cdot π_n$ and in its standard cycle form as $π= (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of pattern avoidance of one-line and the standard cycle form for a cyclic permutation $π$, defined as both $π_1π_2 \cdot\cdot\cdot π_n$ and its standard cycle form $c_1c_2\cdot\cdot\cdot c_{n}$ avoiding a given pattern. Let $\mathcal{A}_n(σ_1,...,σ_k; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid each pattern of $\{σ_1,...,σ_k\}$ in their one-line forms and avoid $τ$ in their standard cycle forms. In this paper, we obtain some results about the cyclic permutations avoiding patterns in both one-line and cycle forms. In particular, we resolve two conjectures of Archer et al.

On two conjectures about pattern avoidance of cyclic permutations

TL;DR

The paper studies pattern avoidance for cyclic permutations in both one-line and standard cycle forms, focusing on the enumerative function . It resolves two Archer et al. conjectures by constructing bijections and structural analyses: (i) for patterns with , , and (ii) . The authors also derive several symmetric and additional enumerative results, including , , , and , as well as a detailed proof converting to via decompositions and size-preserving bijections. These results enhance understanding of pattern avoidance in cyclic permutations and connect to classical sequences such as Fibonacci and Pell numbers.

Abstract

Let be a cyclic permutation that can be expressed in its one-line form as and in its standard cycle form as where . Archer et al. introduced the notion of pattern avoidance of one-line and the standard cycle form for a cyclic permutation , defined as both and its standard cycle form avoiding a given pattern. Let denote the set of cyclic permutations in the symmetric group that avoid each pattern of in their one-line forms and avoid in their standard cycle forms. In this paper, we obtain some results about the cyclic permutations avoiding patterns in both one-line and cycle forms. In particular, we resolve two conjectures of Archer et al.
Paper Structure (5 sections, 19 theorems, 64 equations)

This paper contains 5 sections, 19 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathcal{A}_n(\tau,4321; 213)$
  4. $\mathcal{A}_n(1324,1423; 213)$
  5. Acknowledgement

Key Result

Lemma 2.3

Let $\pi=(1,c_2, ...,c_{j-1},2,3,...,m,m+1)$ and $\pi'=(1,c_2-m, ...,c_{j-1}-m)$, where $2<j\leq n$ and $m=n-j+1$. If $\pi=\pi_13\cdot\cdot\cdot (m+1)1\pi_{m+2}\cdot\cdot\cdot\pi_n$, then Conversely, if $\pi'=(\pi_1-m)(\pi_{m+2}-m)\cdot\cdot\cdot(\pi_{c_{j-1}-1}-m)(\pi_{c_{j-1}}-1)(\pi_{c_{j-1}+1}-m)\cdot\cdot\cdot(\pi_n-m)$, then

Theorems & Definitions (40)

  • Conjecture 1.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • Remark 2.7
  • ...and 30 more