Algebraic Combinatorics of Magic Squares

TL;DR

This paper applies algebraic combinatorics techniques, such as Hilbert series and bases, to construct and enumerate various types of magic squares, cubes, and graphs as lattice points within polyhedral cones.

Contribution

It introduces a novel algebraic combinatorics framework for constructing and enumerating magic structures using Hilbert series and bases.

Findings

Enumeration of magic squares, cubes, and graphs as lattice points

Description of polytopes of magic labelings of graphs and digraphs

Characterization of faces of the Birkhoff polytope

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.