The Statistic $\mathtt{pinv}$ for Number System

TL;DR

This paper introduces a new statistic called the number of pseudoinversions for colored permutation groups, enabling the definition of a novel number system and bijection with positive integers.

Contribution

It generalizes the inversion statistic to colored permutations and constructs a corresponding number system with a bijection to positive integers.

Findings

Defined the statistic number of pseudoinversions.

Constructed a new number system based on this statistic.

Established a bijection between positive integers and colored permutation groups.

Abstract

The number of inversions is a statistic on permutation groups measuring the degree to which the entries of a permutation are out of order. We provide a generalization of that statistic by introducing the statistic number of pseudoinversions on the colored permutation groups. The main motivation to investigate that statistic is the possibility to use it to define a number system and a numeral system on the colored permutation groups. By means of the statistic number of $i$-pseudoinversions, we construct our number system, and a bijection between the set of positive integers and the colored permutation groups.