Kasteleyn Cokernels

We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus matrices. We apply these ideas to plane partitions and related planar of tilings. We list a number of conjectures, supported by experiments in Maple, about the forms of matrices associated to enumerations of plane partitions and other lozenge tilings of planar regions and their symmetry classes. We focus on the case where the enumerations are round or q-round, and we conjecture that cokernels remain round or q-round for related "impossible enumerations" in which there are no tilings. Our conjectures provide a new view of the topic of enumeratin...

• Type: Journal
• Year: 2002
• Where: COMBINATORICS
• Authors: Greg Kuperberg
• Keywords: COMBINATORICS 2002, Enumeration, Kasteleyn-Percus Matrices, Smith Normal Form