Counting of lattices containing up to five comparable reducible elements and having nullity up to three
B. P. Aware, A. N. Bhavale
TL;DR
The paper advances lattice enumeration by solving the counting problem for RC-lattices on $n$ elements with exactly five comparable reducible elements and nullity $3$, building on prior results for fewer reducible elements. It develops a decomposition framework into basic blocks and chains, identifies 30 possible height-configuration blocks $(B_1,\dots,B_{30})$ with heights in $\{4,5,6,7\}$, and provides explicit counting formulas for each height class. Through adjunct representations and block-assembly arguments, it derives detailed sums and closed-form expressions for each class $|\mathscr{B}(j;5,3,h)|$ and finally the total $|\mathscr{L}(n;5,3)|$ for $n\ge 8$. This yields concrete enumeration formulas for five-reducible-element RC-lattices with nullity three, extending the combinatorial toolkit for poset and lattice counting via basic-block decompositions and chain attachments.
Abstract
In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. In 2003 Pawar and Waphare counted all non-isomorphic lattices on n elements and n edges, which are precisely lattices of nullity one. In 2002 Thakare et al. counted all non-isomorphic lattices on n elements containing two reducible elements. In the same paper, Thakare et al. counted lattices on n elements containing up to n+1 edges, which are precisely lattices of nullity up to two. In 2024 Bhavale and Aware counted all non-isomorphic lattices on n elements, containing up to three reducible elements. Recently, Aware and Bhavale counted all non-isomorphic lattices on n elements, containing four comparable reducible elements, and having nullity three. In this paper, we count up to isomorphism the class of all lattices on n elements containing five comparable reducible elements, and having nullity three.