Enumeration of cyclic permutations in vector grid classes

TL;DR

This paper develops a bijective method to enumerate cyclic permutations in vector grid classes, completing the enumeration for length 3 classes and exploring equivalence and properties of these permutations.

Contribution

It introduces a novel bijective approach linking cyclic permutations in vector grid classes to necklaces, enabling complete enumeration for specific cases.

Findings

Complete enumeration of cyclic permutations in length 3 vector grid classes

Established an analog of Wilf-equivalence for these sets

Discussed properties of cyclic permutations in alternating grid classes

Abstract

A grid class consists of permutations whose pictorial depiction can be partitioned into increasing and decreasing parts as determined by a given matrix. In this paper, we introduce a method for enumerating cyclic permutations in vector grid classes by establishing a bijective relationship with certain necklaces. We use this method to complete the enumeration of cyclic permutations in the length 3 vector grid classes. In addition, we define an analog of Wilf-equivalence between these sets. We conclude by discussing cyclic permutations in alternating grid classes.