Counting lattice triangulations: Fredholm equations in combinatorics
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912139627724800 |
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| author | Orevkov, S. Yu. |
| author_facet | Orevkov, S. Yu. |
| contents | Let $f(m,n)$ be the number of primitive lattice triangulations of $m\times n$ rectangle. We compute the limits $\lim_n f(m,n)^{1/n}$ for $m=2$ and $3$. For $m=2$ we obtain the exact value of the limit which is equal to $(611+\sqrt{73})/36$. For $m=3$, we express the limit in terms of certain Fredholm's integral equation on generating functions. This provides a polynomial time algorithm for computation of the limit with any given precision (polynomial with respect the the number of computed digits). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_12827 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Counting lattice triangulations: Fredholm equations in combinatorics Orevkov, S. Yu. Combinatorics Let $f(m,n)$ be the number of primitive lattice triangulations of $m\times n$ rectangle. We compute the limits $\lim_n f(m,n)^{1/n}$ for $m=2$ and $3$. For $m=2$ we obtain the exact value of the limit which is equal to $(611+\sqrt{73})/36$. For $m=3$, we express the limit in terms of certain Fredholm's integral equation on generating functions. This provides a polynomial time algorithm for computation of the limit with any given precision (polynomial with respect the the number of computed digits). |
| title | Counting lattice triangulations: Fredholm equations in combinatorics |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2201.12827 |