Combinatorial identities using Bernoulli Graphs

TL;DR

This paper introduces combinatorial identities derived through a graph-based probabilistic approach, enabling straightforward generalizations to higher dimensions and node counts.

Contribution

It presents a novel graph-based method for deriving combinatorial identities with easy scalability to more complex systems.

Findings

Derived combinatorial equalities using probabilistic graph dynamics

Method allows straightforward generalization to higher dimensions

Probabilistic approach simplifies combinatorial calculations

Abstract

In here, I present a series of combinatorial equalities derived using a graph based approach. Different nodes in the graphs are visited following probabilistic dynamics of a moving dot. The results are presented in such a way that the generalisation (to more nodes, or dimensions) is straightforward. At an instant m, we "take a picture" of the system and we compute the probabilities of being at particular positions in space. The sum of all these probabilities is equal to one.