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λ > 4

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Algorithms - ESA 2015

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9294))

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Abstract

A polyomino (“lattice animal”) is an edge-connected set of squares on the two-dimensional square lattice. Counting polyominoes is an extremely hard problem in enumerative combinatorics, with important applications in statistical physics for modeling processes of percolation and collapse of branched polymers. We investigated a fundamental question related to polyominoes, namely, what is their growth constant, the asymptotic ratio between A(n + 1) and A(n) when n → ∞, where A(n) is the number of polyominoes of size n. This value is also known as “Klarner’s constant” and denoted by λ. So far, the best lower and upper bounds on λ were roughly 3.98 and 4.65, respectively, and so not even a single decimal digit of λ was known. Using extremely high computing resources, we have shown (still rigorously) that λ > 4.00253, thereby settled a long-standing problem: proving that the leading digit of λ is 4.

We acknowledge the support of the facilities and staff of the HPI Future SOC Lab in Potsdam, where this project has been carried out.

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Authors and Affiliations

  1. Dept. of Computer Science, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Gill Barequet & Mira Shalah

  2. Institut für Informatik, Freie Universität Berlin, Takustraße 9, D-14195, Berlin, Germany

    Günter Rote

Authors
  1. Gill Barequet
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  2. Günter Rote
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  3. Mira Shalah
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Correspondence to Gill Barequet .

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Editors and Affiliations

  1. University of Technology, Eindhoven, The Netherlands

    Nikhil Bansal

  2. Sapienza University of Rome, Rome, Italy

    Irene Finocchi

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Barequet, G., Rote, G., Shalah, M. (2015). λ > 4. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_8

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  • DOI: https://doi.org/10.1007/978-3-662-48350-3_8

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  • Online ISBN: 978-3-662-48350-3

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