On the number of antichains of sets in a finite universe

TL;DR

This paper explores the structure of the lattice of antichains of subsets in a finite universe, introducing new concepts, formulas, and methods for analyzing and computing properties of these lattices, including their size and related combinatorial quantities.

Contribution

It introduces new objects, formulas, and decomposition methods for the lattice of antichains, enabling efficient computation of its size and related properties, including the largest known lattice of order 8.

Findings

Derived formulas for the number of connected and fully distinguishing antichains

Established a connection with Stirling numbers of the second kind

Developed an efficient method to compute the size of the lattice of order 8

Abstract

Properties of intervals in the lattice of antichains of subsets of a universe of finite size are investigated. New objects and quantities in this lattice are defined. Expressions and numerical values are deduced for the number of connected antichains and the number of fully distinguishing antichains. The latter establish a connection with Stirling numbers of the second kind. Decomposition properties of intervals in the lattice of antichains are proven. A new operator allowing partitioning the full lattice in intervals derived from lower dimensional sub-lattices is introduced. Special posets underlying an interval of antichains are defined. The poset allows the derivation of a powerful formula for the size of an interval. This formula allows computing intervals in the six dimensional space. Combinatorial coefficients allowing another decomposition of the full lattice are defined. In some specific cases, related to connected components in graphs, these coefficients can be efficiently computed. This formula allows computing the size of the lattice of order 8 efficiently. This size is the number of Dedekind of order 8, the largest one known so far.