# R(p,q)-deformed combinatorics: full characterization and illustration

## TL;DR

This paper develops a comprehensive R(p,q)-deformed combinatorics framework, extending classical combinatorial concepts and applying it to deformed quantum algebra, enriching the mathematical tools for discrete probability and quantum algebra applications.

## Contribution

It provides a full characterization of R(p,q)-deformed combinatorics, including new formulas and properties, and illustrates its application to generalized q-Quesne deformed quantum algebra.

## Key findings

- Derived R(p,q)-deformed factorials and binomial coefficients
- Established R(p,q)-Stirling numbers and Bell numbers
- Applied formalism to q-Quesne deformed quantum algebra

## Abstract

This paper addresses a theory of R(p,q)-deformed combinatorics in discrete probability. It mainly focuses on R(p,q)-deformed factorials, binomial coefficients, Vandermonde's formula, Cauchy's formula, binomial and negative binomial formulae, factorial and binomial moments, and Stirling numbers. Moreover, the R(p,q)-Stirling numbers of the second kind and the R(p,q)-Bell numbers for graphs are also derived. Related relevant properties are investigated and discussed. Finally, as a concrete illustration, the developed formalism is displayed for the well known generalized q-Quesne deformed quantum algebra to construct the corresponding deformed combinatorics, as a particular case.