# Fractional Exclusion Statistics as an Occupancy Process

## TL;DR

This paper models fractional exclusion statistics as an occupancy process with specific constraints, deriving exact rules for state occupation and global occupancy shapes based on combinatorial identities.

## Contribution

It introduces a novel combinatorial framework to describe FES as a ball-in-box occupancy model with explicit exclusion constraints.

## Key findings

- Derived exact occupancy rules for FES systems.
- Established a generalized Pauli principle for FES.
- Identified families of FES systems based on parameters.

## Abstract

We show the possibility of describing fractional exclusion statistics (FES) as an occupancy process with global and \textit{local} exclusion constraints. More specifically, using combinatorial identities, we show that FES can be viewed as "ball-in-box" models with appropriate weighting on the set of occupancy configurations (merely represented by a partition of the total number of particles). As a consequence, the following exact statement of the generalized Pauli principle is derived: for an $N$-particles system exhibiting FES of extended parameter \mbox{$g=q/r$} ($q$ and $r$ are co-prime integers such that $0 < q \leq r$), (1)~the allowed occupation number of a state is less than or equal to $r-q+1$ and \emph{not} to $1/g$ whenever $q\neq 1$ and (2)~the global occupancy shape is admissible if the number of states occupied by at least two particles is less than or equal to $(N-1)/r$ ($N \equiv 1 \mod r$). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter $g$ and the size $N$.