Counting of lattices having nullity up to two

In <span class="math inline">2020</span> 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 <span class="math inline"><em>k</em></span> contains <span class="math inline"><em>r</em></span> reducible elements then <span class="math inline">2 ≤ <em>r</em> ≤ 2<em>k</em></span>. In <span class="math inline">2003</span> Pawar and Waphare counted all non-isomorphic lattices on <span class="math inline"><em>n </em></span>elements having nullity one, containing exactly two reducible elements. Recently, Bhavale and Aware counted all non-isomorphic lattices on <span class="math inline"><em>n</em></span> elements having nullity two, containing up to three reducible elements. In this paper, we count up to isomorphism the class of all lattices on <span class="math inline"><em>n</em></span> elements having nullity two, containing exactly four reducible elements.