A generic method for the enumeration of various classes of directed polycubes

Abstract

Following the track of polyominoes, in particular the column-by-column construction of Temperley and its interpretation in terms of functional equations due to Bousquet-Mélou, we introduce a generic method for the enumeration of classes of directed polycubes the strata of which satisfy some property P. This method is applied to the enumeration of two new families of polycubes, the s-directed polycubes and the vertically-convex s-directed polycubes, with respect to width and volume. The case of non-directed polycubes is also studied and it is shown that the generic method can be applied in this case too. Finally the general case of d-dimensional polycubes, with d ≥ 4, is investigated, and the generic method is extended in order to handle the enumeration of classes of directed d-polycubes. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.