On two conjectures about pattern avoidance of cyclic permutations

TL;DR

This paper investigates pattern avoidance in cyclic permutations considering both one-line and cycle forms, and resolves two conjectures related to these avoidance patterns.

Contribution

It provides new results on pattern avoidance in cyclic permutations and confirms two conjectures posed by Archer et al.

Findings

Characterization of cyclic permutations avoiding specific patterns.

Proof of two conjectures about pattern avoidance in cyclic permutations.

Enhanced understanding of pattern avoidance in both one-line and cycle representations.

Abstract

Let $\pi$ be a cyclic permutation that can be expressed in its one-line form as $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and in its standard cycle form as $\pi = (c_1,c_2, ..., c_n)$ 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 $\pi$, defined as both $\pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and its standard cycle form $c_1c_2\cdot\cdot\cdot c_{n}$ avoiding a given pattern. Let $\mathcal{A}_n(\sigma_1,...,\sigma_k; \tau)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid each pattern of $\{\sigma_1,...,\sigma_k\}$ in their one-line forms and avoid $\tau$ 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.