Counting quadrant walks via Tutte's invariant method

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Paper Summary

Paperzilla title

Tutte's Tricks Tame Treacherous Quadrant Walks (Mostly)

This paper adapts Tutte's invariant method to analyze quadrant walks, proving algebraicity for several models, including Gessel's. It introduces a weaker invariant notion, demonstrating D-algebraicity for nine non-D-finite models with decoupling functions, using a novel integral-free expression for their generating function.

Possible Conflicts of Interest

Supported by NSF and the European Research Council - no obvious conflicts identified.

Identified Weaknesses

Computer algebra dependence

The heavy reliance on computer algebra in some proofs limits accessibility and understanding for those without specialized software.

Limited scope

The focus on small-step quadrant walks leaves open the generalization to larger steps or other lattice path models.

Difficulty with 't' equations

While D-algebraicity is established, explicit differential equations in 't' are noted as more challenging to derive.

Rating Explanation

A strong paper extending Tutte's invariant method to quadrant walks, providing new algebraic and D-algebraic results. The heavy use of computer algebra in some proofs and the limited scope slightly lower the rating.

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Topic Hierarchy

Domain:

Physical Sciences

Field:

Mathematics

Subfield:

Discrete Mathematics and Combinatorics

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Counting quadrant walks via Tutte's invariant method

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