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Enumeration of quarter-plane lattice walks with small steps is a classicalproblem in combinatorics. An effective approach is the kernel method and thesolution is derived by positive term extraction. Another approach is to reducethe lattice walk problem to a Carleman type Riemann boundary value problem(RBVP) and solve it by complex analysis. There are two parameter characterizingthe solutions of a Carleman type Riemann Boundary value problem, the index$\chi$ and the conformal gluing function $w(x)$. In this paper, we propose acombinatorial insight to the RBVP approach. We show that the index can betreated as the canonical factorization in the kernel method and the conformalgluing function can be turned into a conformal mapping such that after thismapping, the positive degree terms and the negative degree terms can benaturally separated. The combinatorial insight of RBVP provide a connectionbetween the kernel method, the RBVP approach and the Tutte's invariant method.

Combinatorial insight of Riemann Boundary value problem in lattice walk problems