The Perimeter of Proper Polycubes
نویسنده
چکیده
We derive formulas for the number of polycubes of size n and perimeter t that are proper in n − 1 and n − 2 dimensions. These formulas complement computer based enumerations of perimeter polynomials in percolation problems. We demonstrate this by computing the perimeter polynomial for n = 12 in arbitrary dimension d.
منابع مشابه
Formulae and Growth Rates of High-Dimensional Polycubes
A d-dimensional polycube is a facet-connected set of cubes in d dimensions. Fixed polycubes are considered distinct if they differ in their shape or orientation. A proper d-dimensional polycube spans all the d dimensions, that is, the convex hull of the centers of its cubes is d-dimensional. In this paper we prove rigorously some (previously conjectured) closed formulae for fixed (proper and im...
متن کاملProper n-Cell Polycubes in n - 3 Dimensions
A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d − 1)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in combinatorics and discrete geometry. This is also an important tool in statistical physics for co...
متن کاملAutomatic Proofs for Formulae Enumerating Proper Polycubes
We develop a general framework for computing formulae enumerating polycubes of size n which are proper in n−k dimensions (spanning all n−k dimensions), for a fixed value of k. Besides the fundamental importance of knowing the number of these simple combinatorial objects, such formulae are central in the literature of statistical physics in the study of percolation processes and the collapse of ...
متن کاملProper n-cell Polycubes in n− k Dimensions
In this paper we develop a theoretical framework for computing the explicit formula enumerating polycubes with n cubes that span n−k dimensions, for a fixed k and variable n. Besides the fundamental importance of knowing the number of these simple combinatorial objects, known as proper polycubes, such formulae are central in the literature of statistical physics in the study of percolation proc...
متن کاملEnumeration of Specific Classes of Polycubes
The aim of this paper is to gather several new results concerning the enumeration of specific classes of polycubes. We first consider two classes of 3-dimensional vertically-convex directed polycubes: the plateau polycubes and the parallelogram polycubes. An expression of the generating function is provided for the former class, as well as an asymptotic result for the number of polycubes of eac...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017