Lattice path enumeration for semi-magic squares by Latin rectangles

TL;DR

This paper explores the enumeration of semi-magic squares using Latin rectangles, analyzing their symmetries, orbits, and chains for sizes 4, 5, and 6, and linking them to Latin squares and hypergraph classes.

Contribution

It introduces a novel approach connecting Latin rectangles to semi-magic square paths and provides complete descriptions for small sizes, including efficient methods for size 6.

Findings

Complete description of orbits, covering data, and chains for sizes 4, 5, and 6.

Identification of hypergraph equivalence classes to compute size 6 cases.

Connection between Latin rectangles, semi-magic squares, and Latin squares.

Abstract

Similar to how standard Young tableaux represent paths in the Young lattice, Latin rectangles may be use to enumerate paths in the poset of semi-magic squares with entries zero or one. The symmetries associated to determinant preserve this poset, and we completely describe the orbits, covering data, and maximal chains for squares of size 4, 5, and 6. The last item gives the number of Latin squares in these cases. To calculate efficiently for size 6, we in turn identify orbits with certain equivalence classes of hypergraphs.