An inequality on broken chessboards

Abstract

For any partition of a positive integer we consider the chess (or draughts) colouring of its associated Ferrers graph. Let b denote the total number of black unit squares, and w the number of white squares. In this note we characterize all pairs (b,w) which arise in this way. This simple combinatorical result was discovered by characterizing Hilbert series of certain right modules over cubic three-dimensional Artin-Schelter algebras. However in this note we present a purely combinatorical proof. The result is (at least partially) known in literature, however we found it interesting to present an elementary proof. All additional references and remarks will be mostly appreciated.