Open AccessPosted Content
On path partitions of the divisor graph
Paul Melotti,Eric Saias +1 more
TLDR
In this article, it was shown that in such a partition, the longest path can have length asymptotically $N^{1-o(1) + 1/n 2/n 1/1/n/n log n/n) in the divisor graph, where n is the number of paths in the graph.Abstract:
It is known that the longest simple path in the divisor graph that uses integers $\leq N$ is of length $\asymp N/\log N$. We study the partitions of $\{1,2,\dots, N\}$ into a minimal number of paths of the divisor graph, and we show that in such a partition, the longest path can have length asymptotically $N^{1-o(1)}$.read more
Citations
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Counting primitive subsets and other statistics of the divisor graph of {1,2,…,n}
TL;DR: A new proof is given that Q ( n ) = α ( 1 + o ( 1 ) ) n for some constant α , which allows for a good error term and to improve upon the lower bound for the value of α.
Dissertation
Integrable spin, vertex and loop models
TL;DR: In this paper, the star-triangle transformation of the Ising model was recast into a singe polynomial evolution equation by Kashaev, and it was shown that this evolution creates combinatorial objects: C2(1) loop models.
Posted Content
Etude du graphe divisoriel 4
Pierre Mazet,Eric Saias +1 more
TL;DR: In this article, Chen and Ji showed that there is a permutation of the positive integers such that for any constant l.c.m.$(f(n), f(n+1)) \leq cn(log n)^2, where lc$ is a positive constant.
References
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Arithmetical properties of permutations of integers
TL;DR: In this article, the least common multiple and the greatest common divisor of two subsequent elements were investigated for the finite and infinite case, respectively, and it was shown that for suitable other permutations this value becomes considerably smaller.
Journal ArticleDOI
Etude Du Graphe Divisoriel 3
TL;DR: In this article, the least cardinal number of elements of a partition of {1, 2, …, N} into simple paths of its divisorial graph was shown to be the smallest cardinal number that is large enough for all n ≥ 1, and the upper bound for n large enough.
Journal ArticleDOI
Recouvrements Hamiltoniens de certains graphes
TL;DR: The results are applied to the particular case of the divisorial graph to give some new answers to a question asked by P. Erdos.