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On a question about pattern avoidance of cyclic permutations

Zuo-Ru Zhang, Hongkuan Zhao

Abstract

Recently, Archer et al.\ studied cyclic permutations that avoid the decreasing pattern $δ_k=k(k-1)\cdots21$ in one-line notation and avoid another pattern $τ$ of length $4$ in all their cycle forms. There are three cases in total to consider, namely, $τ=1324, 1342$ and $1432$. They determined two of them, leaving the case $τ=1432$ as an open question. In this paper, we resolve this case by deriving explicit formulas based on an analysis of the structure of cycle forms and an application of Dilworth's theorem.

On a question about pattern avoidance of cyclic permutations

Abstract

Recently, Archer et al.\ studied cyclic permutations that avoid the decreasing pattern in one-line notation and avoid another pattern of length in all their cycle forms. There are three cases in total to consider, namely, and . They determined two of them, leaving the case as an open question. In this paper, we resolve this case by deriving explicit formulas based on an analysis of the structure of cycle forms and an application of Dilworth's theorem.
Paper Structure (4 sections, 15 theorems, 36 equations)

This paper contains 4 sections, 15 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1.1']}
  3. Proof of Theorem \ref{['th1.2']}
  4. Proof of Theorem \ref{['th1.3']}

Key Result

Theorem 1.1

For $n\ge 5$, there holds $a_n^\circ(\delta_3;1432)=\lfloor \frac{n^2+7}{2} \rfloor-2n$ (OEIS A061925).

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 5 more