Pattern avoidance in permutations and their rotations

Abstract

A rotation of a permutation is a new permutation obtained by moving the first several terms of the permutation to the end of the permutation. A circular permutation is the set of all rotations of a permutation. The enumerations of permutations and circular permutations avoiding patterns of length three and four are well studied. In this paper, we provide exact formulas for the number of permutations whose first $k\geq 2$ rotations all avoid a given pattern of length three, as well as the number of permutations whose first three rotations respectively avoid the rotations of a given pattern of length three. In contrast to permutations and circular permutations avoiding patterns of length three, the Wilf-equivalence classes under study are entirely determined by complements and reverses. We also classify and enumerate permutations whose first two rotations avoid different patterns of length three.

Disclosure

“ome of the results in this article were previously included in the doctoral dissertation of the second author [15]. The authors used SageMath for computational experiments and some of the SageMath code was developed with the help of Google AI tools. M. Yin was supported in part by the Simons Travel Support for Mathematicians Grant 00007227. References [1] K. Archer, E. Borsh, J. Bridges, C. Graves, and M. Jeske, Cyclic permutations avoiding patterns in both one-line and cycle”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T19:03:39.413448Z

Structural counts