Some Quotients of Chain Products are Symmetric Chain Orders

Abstract

Canfield and Mason have conjectured that for all subgroups $G$ of the automorphism group of the Boolean lattice $B_n$ (which can be regarded as the symmetric group $S_n$), the quotient order $B_n/G$ is a symmetric chain order. We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, $B_n/G$ is an SCO whenever $G$ is generated by powers of disjoint cycles. In addition, the Boolean lattice $B_n$ can be replaced by any product of finite chains. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.