On the Number of Maximal Antichains in Boolean Lattices for 𝑛 up to 7
We consider two ways how to compute the number of maximal antichains in the Boolean lattice on 𝑛 elements. The first one is based on full direct enumeration, while the second ones relies on concept lattices or Galois lattices (studied in Formal Concept Analysis, an applied branch of lattice theory) and the Dedekind–McNeil completion of a partial order. The last technique also results in the so-called standard contexts of a (concept) lattice, which gives an alternative representation of maximal antichain lattices as binary relations on meet- and join-irreducible elements. The implemented algorithms are parallelised and openly available on the author’s GitHub page https://github.com/dimachine/NonEquivMACs. All the computational results obtained by the author are listed in their records in the On-Line Encyclopedia of Integer Sequences: https://oeis.org/A326359, https://oeis.org/A326360, and https://oeis.org/A348260. These sequences have their famous counterparts known under the name Dedekind numbers, representing the number of antichains of the Boolean lattice 𝑛 or monotone Boolean functions over 𝑛 variables.