Algebraic Combinatorics of Magic Squares

Abstract

We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.

Ph.D. Thesis

References in corpus (1)

• Residues formulae for volumes and Ehrhart polynomials of convex polytopes

Cited by in corpus (7)

• Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method
• A generalization of product construction of multimagic squares
• Constructing All Magic Squares of Order Three
• Enumerating Magic Distinct Labellings of the Cube
• Enumerating extreme points of the polytopes of stochastic tensors: an optimization approac
• Magic graphs and the faces of the Birkhoff polytope
• Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors