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

<p>In 1940, Birkhoff raised the open problem of computing of all posets/lattices on <span class="math inline">\(n\)</span> elements up to isomorphism for small <span class="math inline">\(n\)</span>. Many authors tried to solve this problem by providing algorithms such as nauty. In 2020, Gebhardt and Tawn given an orderly algorithm for constructing unlabelled lattices of given size and explicitly obtained the number of lattices on up to <span class="math inline">\(20\)</span> elements. In 2020, Bhavale and Waphare introduced the concept of nullity of a poset as the nullity of its cover graph. Recently, Bhavale and Aware counted lattices having nullity up to two. Bhavale and Aware also counted all non-isomorphic lattices on <span class="math inline">\(n\)</span> elements, containing up to three reducible elements, having arbitrary nullity <span class="math inline">\(k \geq 2\)</span>. In this paper, we count up to isomorphism the class of all lattices on <span class="math inline">\(n\)</span> elements containing four comparable reducible elements, and having nullity three.</p>