Combinatorial Insight of Riemann Boundary Value Problem in Lattice Walk Problems

TL;DR

This paper reveals a combinatorial connection between the Riemann boundary value problem approach and the kernel method in enumerating quarter-plane lattice walks, unifying different analytical techniques.

Contribution

It introduces a combinatorial perspective that links the RBVP approach with the kernel method and Tutte's invariants in lattice walk enumeration.

Findings

The index in RBVP corresponds to the kernel method's canonical factorization.

The conformal gluing function acts as a mapping facilitating positive term extraction.

Establishes a unifying framework connecting multiple analytical methods in lattice walk enumeration.

Abstract

The enumeration of quarter-plane lattice walks with small steps is a classical problem in combinatorics. An effective approach is the kernel method, where the solution is derived by positive term extraction. Alternatively, one may reduce the lattice walk problem to a Carleman-type Riemann boundary value problem (RBVP) and solve it via analytic method. In the RBVP framework, two parameters govern the solution: the index $\chi$ the conformal gluing function $w(x)$. In this paper, we propose a combinatorial insight into the RBVP approach. We show that the index corresponds to the canonical factorization in the kernel method. The conformal gluing function can be viewed as a mapping that enables the application of positive term extraction. The combinatorial insight of RBVP establishes a unifying link between the kernel method, the RBVP approach and the Tutte's invariants method.