On Hyperoctahedral Enumeration System, Application to Signed Permutations

TL;DR

This paper explores the hyperoctahedral number system and its application to efficiently encoding, ordering, and retrieving signed permutations without lexicographical listing, enhancing combinatorial enumeration methods.

Contribution

It introduces a novel correspondence between hyperoctahedral numbers and signed permutations, enabling direct computation of permutations from their indices.

Findings

Established a correspondence between hyperoctahedral numbers and signed permutations

Developed an ordering scheme for signed permutations based on this correspondence

Enabled direct computation of the r-th signed permutation from its index

Abstract

In this paper, we start by giving the definitions and basic facts about hyperoctahedral number system. There is a natural correspondence between the integers expressed in the latter and the elements of the hyperoctahedral group when we use the inversion statistic on this group to code the signed permutations. We use this correspondence to define a way with which the signed permutations group can be ordered. With this classification scheme, we can find the r-th signed permutation from a given number r and vice versa without consulting the list in lexicographical order of the elements of the signed permutations group.