Pattern-restricted cyclic permutations with a pattern-restricted cycle form

TL;DR

This paper studies cyclic permutations avoiding certain patterns, providing generating functions and enumeration results for cycles with specific pattern restrictions, especially focusing on pattern avoidance in cycle forms and rotations.

Contribution

It introduces new enumeration formulas and generating functions for pattern-restricted cyclic permutations with specific avoidance properties, including continued fractions.

Findings

Derived continued fraction generating functions for pattern-avoiding cycles.

Enumerated classes of cycles avoiding patterns of length 3 and 4.

Provided explicit counts for certain pattern-restricted cycle forms.

Abstract

In this paper, we consider cyclic permutations that avoid the monotone decreasing permutation $k(k-1)\ldots 21$, whose cycle also demonstrates some pattern avoidance. If the cycle is written in standard form with 1 appearing at the beginning of the cycle and the standard form avoids a pattern of length 3, we find answers in terms of continued fraction generating functions. We also consider the case that every cyclic rotation of the cycle form of the permutation avoids a pattern of length 4 and enumerate two such cases.