Extending the symbolic method in enumerative combinatorics. I

2025/11/02 by Martin Klazar, Klazar, M., Richard Horský +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2511.00914

openalex publication_date 2025/11/02 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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 Pólya's theorem. This theorem determines limits of probabilities that walks in the grid graph ℤ^d, starting at the origin, visit the given vertex. We extend ℤ^d to the countable complete graph K_ℕ with weighted edges.

Citations

• Extending Pólya's random walker beyond probability I. Complex weights
• Combinatorial Exploration: An algorithmic framework for enumeration
• Linear Differential Equations as a Data-Structure
• An easy proof of Polya's theorem on random walks
• Polya's random walk theorem
• Random Walks and Electric Networks

Related

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• On self-avoiding polygons and walks: the snake method via polygon joining
• A Combinatorial Lemma for Complex Numbers
• On self-avoiding polygons and walks: counting, joining and closing
• On the Expansion of Graphs of 0/1-Polytopes