Enumeration of Colored Tilings on Graphs via Generating Functions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917889176502272 |
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| author | Ramírez, José L. Villamizar, Diego |
| author_facet | Ramírez, José L. Villamizar, Diego |
| contents | In this paper, we study the problem of partitioning a graph into connected and colored components called blocks. Using bivariate generating functions and combinatorial techniques, we determine the expected number of blocks when the vertices of a graph $G$, for $G$ in certain families of graphs, are colored uniformly and independently. Special emphasis is placed on graphs of the form $G \times P_n$, where $P_n$ is the path graph on $n$ vertices. This case serves as a generalization of the problem of enumerating the number of tilings of an $m \times n$ grid using colored polyominoes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration of Colored Tilings on Graphs via Generating Functions Ramírez, José L. Villamizar, Diego Combinatorics 05A15, 05A05 In this paper, we study the problem of partitioning a graph into connected and colored components called blocks. Using bivariate generating functions and combinatorial techniques, we determine the expected number of blocks when the vertices of a graph $G$, for $G$ in certain families of graphs, are colored uniformly and independently. Special emphasis is placed on graphs of the form $G \times P_n$, where $P_n$ is the path graph on $n$ vertices. This case serves as a generalization of the problem of enumerating the number of tilings of an $m \times n$ grid using colored polyominoes. |
| title | Enumeration of Colored Tilings on Graphs via Generating Functions |
| topic | Combinatorics 05A15, 05A05 |
| url | https://arxiv.org/abs/2501.06008 |