@inproceedings{879bba8a6c664f50be47dc404f892460,

title = "λ > 4",

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{\textquoteright}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.",

keywords = "Growth constant, Lattice animals, Polyominoes",

author = "Gill Barequet and G{\"u}nter Rote and Mira Shalah",

note = "Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2015.; 23rd European Symposium on Algorithms, ESA 2015 ; Conference date: 14-09-2015 Through 16-09-2015",

year = "2015",

doi = "https://doi.org/10.1007/978-3-662-48350-3_8",

language = "الإنجليزيّة",

isbn = "9783662483497",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

pages = "83--94",

editor = "Nikhil Bansal and Irene Finocchi",

booktitle = "Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings",

}