An explicit 1-factorization in the middle of the Boolean lattice
Let \(R_ j\) be the collection of \(j\)-element subsets of \(\{1,2, \dots, 2k+1\}\), and let \(B_ k\) be the bipartite graph defined on \(R_ k \cup R_{k+1}\) by letting \(A\) be adjacent to \(B\) if and only if \(A \subset B\) or vice versa. The authors define explicitly a 1-factorization of \(B_ k\) whose constituent matchings are defined using addition mod \(k+1\). They show that the matchings are invariant under rotation, they describe the effect of reflection, they show that there are no other symmetries that map these matchings among themselves, and they prove that they are distinct from the lexical matchings in \(B_ k\) (for lexical factorization see [\textit{H. A. Kierstead} and \textit{W. T. Trotter}, Explicit matchings in the middle levels of the Boolean lattice, Order 5, No. 2, 163-171 (1988; Zbl 0668.05045)]).
scientific article; zbMATH DE number 554407