A generalization of rotation of binary sequences and its applications to toggle dynamical systems

TL;DR

This paper introduces a generalized rotation operation on binary sequences, revealing statistical symmetries in its orbits, and connects it to toggle dynamical systems on path graph independent sets.

Contribution

It generalizes the rotation of binary sequences and links this to toggle dynamical systems, expanding understanding of their statistical and combinatorial properties.

Findings

Each orbit exhibits a specific statistical symmetry.

The generalized rotation arises naturally in toggle dynamical systems.

Connections to independent sets on path graphs are established.

Abstract

We study a simple generalization of the rotation (or circular shift) of the binary sequences. In particular, we show each orbit of this generalized rotation has a certain statistical symmetry. This generalized rotation naturally arises when we generalize the results of Joseph and Roby on a toggle dynamical system whose state space consists of independent sets on the path graphs.