Automatic Proofs for Formulae Enumerating Proper Polycubes

Gill Barequet; Mira Shalah

doi:10.1016/j.endm.2015.06.022

Automatic Proofs for Formulae Enumerating Proper Polycubes

Gill Barequet, Mira Shalah

Computer Science

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

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 branched polymers. We re-affirm the already-proven formulae for k≤3, and prove rigorously, for the first time, that the number of polycubes of size n that are proper in n-4 dimensions is 2^n-7n^n-9(n-4)(8n⁸-128n⁷+828n⁶-2930n⁵+7404n⁴-17523n³+41527n²-114302n+204960)/6.

Original language	English
Pages (from-to)	145-151
Number of pages	7
Journal	Electronic Notes in Discrete Mathematics
Volume	49
DOIs	https://doi.org/10.1016/j.endm.2015.06.022
State	Published - Nov 2015

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1016/j.endm.2015.06.022

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title = "Automatic Proofs for Formulae Enumerating Proper Polycubes",

abstract = "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 branched polymers. We re-affirm the already-proven formulae for k≤3, and prove rigorously, for the first time, that the number of polycubes of size n that are proper in n-4 dimensions is 2n-7nn-9(n-4)(8n8-128n7+828n6-2930n5+7404n4-17523n3+41527n2-114302n+204960)/6.",

author = "Gill Barequet and Mira Shalah",

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year = "2015",

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doi = "10.1016/j.endm.2015.06.022",

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T1 - Automatic Proofs for Formulae Enumerating Proper Polycubes

AU - Barequet, Gill

AU - Shalah, Mira

PY - 2015/11

Y1 - 2015/11

N2 - 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 branched polymers. We re-affirm the already-proven formulae for k≤3, and prove rigorously, for the first time, that the number of polycubes of size n that are proper in n-4 dimensions is 2n-7nn-9(n-4)(8n8-128n7+828n6-2930n5+7404n4-17523n3+41527n2-114302n+204960)/6.

AB - 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 branched polymers. We re-affirm the already-proven formulae for k≤3, and prove rigorously, for the first time, that the number of polycubes of size n that are proper in n-4 dimensions is 2n-7nn-9(n-4)(8n8-128n7+828n6-2930n5+7404n4-17523n3+41527n2-114302n+204960)/6.

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U2 - 10.1016/j.endm.2015.06.022

DO - 10.1016/j.endm.2015.06.022

M3 - مقالة

SN - 1571-0653

VL - 49

SP - 145

EP - 151

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

ER -

Automatic Proofs for Formulae Enumerating Proper Polycubes

Abstract

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