A generic method for the enumeration of various classes of directed polycubes
Jean-Marc Champarnaud, Jean-Philippe Dubernard, Hadrien Jeanne
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.
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