Extending the symbolic method in enumerative combinatorics. I

TL;DR

This paper extends the symbolic method in enumerative combinatorics to infinite series, generalizing Pólya's theorem to analyze visit probabilities in extended graph structures.

Contribution

It introduces a novel extension of the symbolic method to infinite sums, enabling the generalization of Pólya's theorem for complex graph structures.

Findings

Generalized Pólya's theorem for infinite graphs

Extended symbolic method to infinite series

Analyzed visit probabilities in weighted countable graphs

Abstract

We use our extension of the symbolic method in enumerative combinatorics (we extend finite sums defining coefficients in generating functions to infinite series) to generalize P\'olya's theorem. This theorem determines limits of probabilities that walks in the grid graph $\mathbb{Z}^d$, starting at the origin, visit the given vertex. We extend $\mathbb{Z}^d$ to the countable complete graph $K_{\mathbb{N}}$ with weighted edges.