Automatic Proofs for Formulae Enumerating Proper Polycubes

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• 作者：Gill Barequet ^{barequet@cs.technion.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Electronic Notes in Discrete Mathematics
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：49
• 期：Complete
• 页码：145-151
• 全文大小：272 K

文摘

We develop a general framework for computing formulae enumerating polycubes of size n   which are proper in e1682cb73" title="Click to view the MathML source">n−k$n-k$ dimensions (spanning all e1682cb73" title="Click to view the MathML source">n−k$n-k$ 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 branched polymers. We re-affirm the already-proven formulae for k≤3$k\le 3$, and prove rigorously, for the first time, that the number of polycubes of size n   that are proper in n−4$n-4$ dimensions is 2ⁿ⁻⁷nⁿ⁻⁹(n−4)(8n⁸−128n⁷+828n⁶−2930n⁵+7404n⁴−17523n³+41527n²−114302n+204960)/6$2^{n-7}{n}^{n-9}(n-4)(8n^8-128n^7+828n^6-2930n^5+7404n^4-17523n^3+41527n^2-114302n+204960)/6$.