Proper n-cell polycubes in n - 3 dimensions

Andrei Asinowski; Gill Barequet; Ronnie Barequet; Günter Rote

doi:10.1007/978-3-642-22685-4_16

Proper n-cell polycubes in n - 3 dimensions

Andrei Asinowski
, Gill Barequet
, Ronnie Barequet
, Günter Rote

Computer Science

Technion - Israel Institute of Technology

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

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 discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in (n - 3) dimensions.

Original language	English
Title of host publication	Computing and Combinatorics - 17th Annual International Conference, COCOON 2011, Proceedings
Pages	180-191
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-642-22685-4_16
State	Published - 2011
Event	17th Annual International Computing and Combinatorics Conference, COCOON 2011 - Dallas, TX, United States Duration: 14 Aug 2011 → 16 Aug 2011

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6842 LNCS

Conference

Conference	17th Annual International Computing and Combinatorics Conference, COCOON 2011
Country/Territory	United States
City	Dallas, TX
Period	14/08/11 → 16/08/11

Keywords

Lattice animals
directed trees
polyominoes

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-642-22685-4_16

Cite this

Asinowski, A., Barequet, G., Barequet, R., & Rote, G. (2011). Proper n-cell polycubes in n - 3 dimensions. In Computing and Combinatorics - 17th Annual International Conference, COCOON 2011, Proceedings (pp. 180-191). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6842 LNCS). https://doi.org/10.1007/978-3-642-22685-4_16

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abstract = "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 discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in (n - 3) dimensions.",

keywords = "Lattice animals, directed trees, polyominoes",

author = "Andrei Asinowski and Gill Barequet and Ronnie Barequet and G{\"u}nter Rote",

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Asinowski, A, Barequet, G, Barequet, R & Rote, G 2011, Proper n-cell polycubes in n - 3 dimensions. in Computing and Combinatorics - 17th Annual International Conference, COCOON 2011, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6842 LNCS, pp. 180-191, 17th Annual International Computing and Combinatorics Conference, COCOON 2011, Dallas, TX, United States, 14/08/11. https://doi.org/10.1007/978-3-642-22685-4_16

Proper n-cell polycubes in n - 3 dimensions. / Asinowski, Andrei; Barequet, Gill; Barequet, Ronnie et al.
Computing and Combinatorics - 17th Annual International Conference, COCOON 2011, Proceedings. 2011. p. 180-191 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6842 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - 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 discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in (n - 3) dimensions.

AB - 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 discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in (n - 3) dimensions.

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KW - directed trees

KW - polyominoes

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Asinowski A, Barequet G, Barequet R, Rote G. Proper n-cell polycubes in n - 3 dimensions. In Computing and Combinatorics - 17th Annual International Conference, COCOON 2011, Proceedings. 2011. p. 180-191. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-22685-4_16

Proper n-cell polycubes in n - 3 dimensions

Abstract

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Conference

Keywords

ASJC Scopus subject areas

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