Counting graded lattices of rank three that have few coatoms

TL;DR

This paper develops a computational approach to count unlabeled graded lattices of rank 3 with a limited number of coatoms and atoms, providing exact counts and quasipolynomial formulas for small c values.

Contribution

It introduces a method combining constructive enumeration and combinatorics to compute R(c,a) for small c and large a, and derives explicit quasipolynomial formulas for c ≤ 7.

Findings

Exact values of R(c,a) for c ≤ 9 and a ≤ 1000

Existence of quasipolynomials matching R(c,a) for large a

Closed form expressions for c ≤ 7

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 $c \le 7$.