Counting graded lattices of rank 3 that have few coatoms

Abstract

We consider the problem of computing [Formula: see text], the number of unlabeled graded lattices of rank [Formula: see text] that contain [Formula: see text] coatoms and [Formula: see text] atoms. More specifically, we do this when [Formula: see text] is fairly small, but [Formula: see text] 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 [Formula: see text] for [Formula: see text] and [Formula: see text]. We also show that, for any fixed [Formula: see text], there exists a quasipolynomial in [Formula: see text] that matches with [Formula: see text] for all [Formula: see text] above a small value. We explicitly determine these quasipolynomials for [Formula: see text], thus finding closed form expressions of [Formula: see text] for [Formula: see text].