    ## 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

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 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\$.