A Reciprocity Theorem for Monomer-Dimer Coverings

The problem of counting the number of monomer-dimer coverings of a lattice comes from the field of statistical mechanics. It has only been shown to be exactly solved for the special case of dimer coverings in two dimensions ([2], [9]). In earlier work, Stanley [8] proved a reciprocity principle governing the number N(m, n) of dimer coverings of an m by n rectangular grid (also known as perfect matchings), where m is fixed and n is allowed to vary. As reinterpreted by Propp [5], Stanley’s result concerns the unique way of extending N(m, n) to n < 0 so that the resulting bi-infinite sequence, N(m, n) for n ∈ Z, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that N(m, n) is always an integer satisfying the relation N(m, −2 − n) = m,nN(m, n) where m,n = 1 unless

• Type: Conference
• Year: 2003
• Where: DMCS
• Authors: Nick Anzalone, John Baldwin, Ilya Bronshtein, T. Kyle Petersen
• Keywords: Dimer Coverings, DMCS 2003, Linear Recurrence Relation, Modeling And Simulation, Monomer-dimer Coverings