Counting graded lattices of rank three that have few coatoms. (arXiv:1804.03679v2 [math.CO] UPDATED)

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. We describe a computational method that, for a fixed $c$, combines direct enumeration of the connection graphs of $c$ coatoms, and Redfield--P\'olya counting for distributing atoms between the coatoms. Using this method we compute $R(c,a)$ for $c\le 9$ and $a\le 1000$. With the help of these computations we also derive $R(c,a)$ in closed form for $c \le 7$. 查看全文>>