Intervals of Antichains and Their Decompositions
Patrick De Causmaecker, Stefan De Wannemacker, Jay Yellen
TL;DR
This paper explores the structure of antichains within the lattice of subsets, providing new partitioning theorems and formulas for counting intervals, advancing understanding of combinatorial set systems.
Contribution
It introduces novel partitioning theorems and counting formulas for intervals in the lattice of antichains, enriching combinatorial theory.
Findings
Derived new partitioning theorems for antichain intervals
Established formulas for counting the size of intervals
Enhanced understanding of the lattice structure of antichains
Abstract
An antichain of subsets is a set of subsets such that no subset in the antichain is a proper subset of any other subset in the antichain. The Dedekind number counts the total number of antichains of subsets of an n-element set. This paper investigates the interval structure of the lattice of antichains. Several partitioning theorems and counting formulas for the size of intervals are derived.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Mathematical Dynamics and Fractals · Advanced Mathematical Identities