Counting graded lattices of rank three that have few coatoms
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 $c \le 7$.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2018
- DOI:
- 10.48550/arXiv.1804.03679
- arXiv:
- arXiv:1804.03679
- Bibcode:
- 2018arXiv180403679K
- Keywords:
-
- Mathematics - Combinatorics;
- 06B99 (Primary) 05C30;
- 20B40 (Secondary)
- E-Print:
- Revised text, added example, corrections