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Bin Decompositions

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 Added by Pamela Harris
 Publication date 2018
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and research's language is English




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It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy $F_n =F_{n-1}+F_{n-2}$ for $ngeq 3$, $F_1 =1$ and $F_2 =2$. In this paper, for any $n,minmathbb{N}$ we create a sequence called the $(n,m)$-bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the $(n,m)$-bin legal decompositions displays Gaussian behavior.



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We describe a new algorithm, the $(k,ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special inst ances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the $(k,ell)$-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and Westermann and Hendrickson.
We describe a new algorithm, the $(k,\\ell)$-pebble game with colors, and use\nit obtain a characterization of the family of $(k,\\ell)$-sparse graphs and\nalgorithmic solutions to a family of problems concerning tree decompositions of\ngraphs. Spe cial instances of sparse graphs appear in rigidity theory and have\nreceived increased attention in recent years. In particular, our colored\npebbles generalize and strengthen the previous results of Lee and Streinu and\ngive a new proof of the Tutte-Nash-Williams characterization of arboricity. We\nalso present a new decomposition that certifies sparsity based on the\n$(k,\\ell)$-pebble game with colors. Our work also exposes connections between\npebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and\nWestermann and Hendrickson.\n
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