Equivalence Classes of Matchings and Lattice-Square
William Y. C. Chen and David C. Torney
Abstract: Nonisomorphic lattice-square designs yielded by a conventional construction are enumerated. Constructed designs are specied by words composed from nite-eld elements. These words are permuted by the isomorphism group in question. The latter group contains a direct product subgroup, acting, respectively, upon the positions and identities of the finite-eld elements. We describe enumeration theory for such direct-product groups. This subgroup is a direct product of a hyperoctahedral and a dihedral group, with the orbits of the hyperoctahedral group, acting on the positions of the eld elements to generate orbits, interpretable as perfect matchings. Thus, the enumeration of dihedral equivalence classes of perfect matchings provides an upper bound on the number of nonisomorphic, constructed designs. The full isomorphism group also contains non-direct-product elements, and the isomorphism classes are enumerated using Burnside's Lemma: counting the number of orbits of a normal subgroup fixed by the quotient group. This approach is applied to constructed lattice-square designs of odd, prime-power order ≤ 13.
Keywords: collineation, combinatorial enumeration, design isomorphism, dihedral group, equivalence class, finite eld, group action, hyperoctahedral group, isomorphism group, linear algebra, semidirect product, spread.