Geodesic
Counting lattice triangulations: Fredholm equations in combinatorics
Sbornik. Mathematics, Tome 213 (2022) no. 11, pp. 1530-1558

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Let $f(m,n)$ be the number of primitive lattice triangulations of an $m\times n$ rectangle. We compute the limits $\lim_n f(m,n)^{1/n}$ for $m=2,3$. For $m=2$ we obtain the exact value of the limit, which is $(611+\sqrt{73})/36$. For $m=3$ we express the limit in terms of a certain Fredholm integral equation for generating functions. This provides a polynomial-time algorithm (with respect to the number of computed digits) for the computation of the limit with any prescribed precision. Bibliography: 13 titles.
Keywords: asymptotic, primitive triangulation, Fredholm's integral equation. 
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     author = {S. Yu. Orevkov},
     title = {Counting lattice triangulations: {Fredholm} equations in combinatorics},
     journal = {Sbornik. Mathematics},
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     publisher = {mathdoc},
     volume = {213},
     number = {11},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2022_213_11_a4/}
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S. Yu. Orevkov. Counting lattice triangulations: Fredholm equations in combinatorics. Sbornik. Mathematics, Tome 213 (2022) no. 11, pp. 1530-1558. http://geodesic.mathdoc.fr/item/SM_2022_213_11_a4/