Generalized Number Systems and Application to Hyperoctahedral Groups

TL;DR

This paper introduces a generalized number system, constructs bijections with specific groups including hyperoctohedral groups, and develops a Lehmer-like code for these groups, advancing algebraic combinatorics.

Contribution

It generalizes integer enumeration bases and creates a new coding scheme for hyperoctohedral groups, which is novel in algebraic combinatorics.

Findings

Generalized integer enumeration basis.

Bijections between special sets and groups.

A Lehmer-like code for hyperoctohedral groups.

Abstract

In this work, we generalize the integer enumeration basis. We also construct bijections between the elements of special sets and the elements of some groups, and treat the special case of the hyperoctohedral groups. Then, we find a code analogous to the Lehmer code for the hyperoctahedral groups.