An Explicit 1-Factorization in the Middle of the Boolean Lattice

An explicit definition of a 1-factorization of B k (the bipartite graph defined by the kand (k+l)-element subsets of [ 2k+ l ] ) , whose constituent matchings are defined using addition modulo k + 1, is introduced. We show that the matchings are invariant under rotation (mapping under ~r = (1, 2, 3 ..... 2k + 1 )), describe the effect of reflection (mapping under p = (1, 2k + 1)(2, 2k).. . (k, k + 2)), determine that there are no other symmetries which map these matchings among themselves, and prove that they are distinct from the lexical matchings in B k. © 1994 Academic Press, Inc.