Combinatorial Insight of Riemann Boundary Value Problem in Lattice Walk Problems

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.