Counting of lattices containing up to four comparable reducible elements and having nullity up to three
B. P. Aware, A. N. Bhavale
TL;DR
This paper completes the enumeration of RC-lattices on $n$ elements with exactly four comparable reducible elements and nullity $k=3$ by classifying all feasible base blocks (22 in total) of height 3–6 and counting their contributions. It extends prior work that solved the $k=2$ case and uses adjunct constructions with chains to build full lattices from maximal blocks, yielding explicit multi-sum formulas for each height class and an overall formula $|\mathscr{L}(n;4,3)|=\sum_{i=0}^{n-7}(i+1)|\mathscr{B}(n-i;4,3)|$. The methodology relies on dissecting the block structure $\mathscr{B}(j;4,3)$ into height-specific subclasses and leveraging partitions $P_n^k$ to enumerate arrangements, ultimately providing a complete enumeration framework for this combinatorial lattice class. The results contribute a concrete, algorithmic pathway for counting lattices with fixed reducible-element structure and nullity, enabling further progress toward comprehensive lattice enumeration by increasing reducible-element counts or nullity benchmarks.
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.