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arXiv · 1804.03679

Counting graded lattices of rank three that have few coatoms

Jukka Kohonen
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Abstract

We consider the problem of computing $R(c,a)$, the number of unlabeled graded lattices of rank $3$ that contain $c$ coatoms and $a$ atoms. More specifically we do this when $c$ is fairly small, but $a$ may be large. For this task, we describe a computational method that combines constructive listing of basic cases and tools from enumerative combinatorics. With this method we compute the exact values of $R(c,a)$ for $c\le 9$ and $a\le 1000$. We also show that, for any fixed $c$, there exists a quasipolynomial in $a$ that matches with $R(c,a)$ for all $a$ above a small value. We explicitly determine these quasipolynomials for $c \le 7$, thus finding closed form expressions of $R(c,a)$ for $c \le 7$.

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Jukka Kohonen. 2018-09-11. Counting graded lattices of rank three that have few coatoms. https://doi.org/10.1142/s0218196719500206

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