Combinatorial Insight of Riemann Boundary Value Problem in Lattice Walk Problems
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916727230562304 |
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| author | Xu, Ruijie |
| author_facet | Xu, Ruijie |
| contents | 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 $χ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11960 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Combinatorial Insight of Riemann Boundary Value Problem in Lattice Walk Problems Xu, Ruijie Combinatorics 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 $χ$ 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. |
| title | Combinatorial Insight of Riemann Boundary Value Problem in Lattice Walk Problems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2308.11960 |