Counting of lattices containing up to four comparable reducible elements and having nullity up to three

TL;DR

This paper classifies specific lattices with up to four comparable reducible elements and nullity three, extending previous counts of lattices with lower nullity or fewer reducible elements.

Contribution

It provides a comprehensive enumeration of lattices with four comparable reducible elements and nullity three, a novel extension of prior lattice counting results.

Findings

Counted all such lattices up to isomorphism

Extended previous nullity and reducibility classifications

Provided explicit enumeration for these lattice classes

Abstract

In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. According to Bhavale and Waphare, if a dismantlable lattice of nullity k contains r reducible elements then 2 $\leq$ r $\leq$ 2k. In 2003 Pawar and Waphare counted all non-isomorphic lattices with equal number of elements and edges, which are precisely the lattices of nullity one. Recently, Bhavale and Aware counted all non-isomorphic lattices on n elements having nullity up to two. Bhavale and Aware also counted all non-isomorphic lattices on n elements, containing up to three reducible elements, having nullity k $\geq$ 2. In this paper, we count up to isomorphism the class of all lattices on n elements containing four comparable reducible elements, and having nullity three.