Counting Near-Perfect Matchings on Cm × Cn Tori of Odd Order in the Maple System

Abstract

In the Maple computer algebra system, a set of recurrence relations and associated generating functions is derived for the number of near-perfect matchings on \({{C}_{m}} \times {{C}_{n}}\) tori of odd order at fixed values of the parameter m (\(3 \leqslant m \leqslant 11\)). The identity of the recurrence relations for the number of perfect and near-perfect matchings is revealed for the same value of m. An estimate for the number of near-perfect matchings is obtained at large odd m when \(n \to \infty \).