Partitions into chains of a class of partially ordered sets

Abstract

Let a cube of side k in R n {{\mathbf {R}}^n} be dissected into k n {k^n} unit cubes. The collection of all affine subspaces of R n {{\mathbf {R}}^n} determined by the faces of the unit cubes forms a lattice L ( n , k ) L(n,k) when ordered by inclusion. We explicitly construct a Dilworth partition into chains of L ( n , k ) L(n,k) .