# Reverse juggling processes

## TL;DR

This paper generalizes reverse juggling Markov chains by introducing weighted variants, resulting in new combinatorial formulas for their stationary distributions, including a multivariate inversion polynomial for permutations.

## Contribution

It extends existing reverse juggling Markov chains with weights, providing explicit stationary distributions and a novel multivariate inversion polynomial for permutations.

## Key findings

- Weighted chains maintain simple stationary distributions
- Derived a multivariate inversion polynomial for permutations
- Generalized Markov chains with combinatorial expressions

## Abstract

Knutson introduced two families of reverse juggling Markov chains (single and multispecies) motivated by the study of random semi-infinite matrices over $\mathbb{F}_q$. We present natural generalizations of both chains by placing generic weights that still lead to simple combinatorial expressions for the stationary distribution. For permutations, this is a seemingly new multivariate generalization of the inversion polynomial.