On the enumeration of k-omino towers

We describe a class of geometric objects called k-omino towers that are created by stacking blocks of size k×1×1 on a convex base composed of these same k-omino blocks. These towers are equivalent to a class of fixed polyominoes or lattice animals, through the creation of a polyomino using the boundary of the two-dimensional vertical face of the tower. By applying a partition to the set of k-omino towers of fixed area kn, we give a recurrence using towers consisting of one less k-omino and bases of varying lengths, therefore showing the set of k-omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity with parameters similar to those given in a classical result of Kummer.