Counting n-cell polycubes proper in n−k dimensions
| Autor: | Mira Shalah, Gill Barequet |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Connected space Conjecture Span (category theory) Value (computer science) Collapse (topology) 0102 computer and information sciences 02 engineering and technology Polycube 01 natural sciences Combinatorics 010201 computation theory & mathematics Simple (abstract algebra) 020204 information systems Percolation 0202 electrical engineering electronic engineering information engineering Discrete Mathematics and Combinatorics Mathematics |
| Zdroj: | European Journal of Combinatorics. 63:146-163 |
| ISSN: | 0195-6698 |
| DOI: | 10.1016/j.ejc.2017.03.006 |
| Popis: | 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. In this paper, we develop a theoretical framework for computing the explicit formula enumerating polycubes of size n that span n − k dimensions, for a fixed value of k . Besides the interest in the number of these simple combinatorial objects, known as proper polycubes, such formulae play an important role in the literature of statistical physics in the study of percolation processes and collapse of branched polymers. The main contribution of this framework is that it enabled us to prove a conjecture about the general form of the formula for a general k . We also used this framework for implementing a computer program which reaffirmed the known formulae for k = 2 and k = 3 , and proved rigorously, for the first time, the formulae for k = 4 and k = 5 . |
| Databáze: | OpenAIRE |
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