The Number System of the Permutations Generated by Cyclic Shift

TL;DR

This paper introduces a number system coding for permutations generated by cyclic shifts, enabling efficient ranking and unranking, and explores its symmetry properties and potential Gray code structure.

Contribution

It proposes a novel coding scheme for cyclic shift permutations, facilitating ranking, unranking, and revealing their symmetry and Gray code conjecture.

Findings

Defines a code for cyclic shift permutation set

Enables ranking and unranking of permutations

Conjectures a Gray code structure for the set

Abstract

A number system coding for the permutations generated by cyclic shift is described. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code describing the symmetry properties of the set of permutations generated by cyclic shift. This code is conjectured to be a combinatorial Gray code listing the set of permutations: this corresponds to an Hamiltonian path of minimal weight in an appropriate regular digraph.