# A Combinatorial Approach to the Opposite Bi-Free Partial $S$-Transform

## TL;DR

This paper introduces a combinatorial method for the opposite bi-free partial $S$-transform, extending it to conditional and operator-valued bi-free contexts, advancing the mathematical understanding of bi-free probability.

## Contribution

It presents a novel combinatorial approach to the opposite bi-free partial $S$-transform and extends it to new settings such as conditional and operator-valued bi-free frameworks.

## Key findings

- Developed a combinatorial framework for the opposite bi-free partial $S$-transform.
- Extended the transform to conditional bi-free probability.
- Extended the transform to operator-valued bi-free probability.

## Abstract

In this paper, we present a combinatorial approach to the opposite 2-variable bi-free partial $S$-transforms where the opposite multiplication is used on the right. In addition, extensions of this partial $S$-transforms to the conditional bi-free and operator-valued bi-free settings are discussed.