منابع مشابه
Polytopes of partitions of numbers
We study the vertices and facets of the polytopes of partitions of numbers. The partition polytope n P is the convex hull of the set of incidence vectors of all partitions 1 2 2 ... n n x x nx = + + +. We show that the sequence 1 2 n P P P can be treated as an embedded chain. Dynamics of behavior of the vertices of n P , as n increases, is established. Some sufficient and some necessary conditi...
متن کاملCayley Compositions, Partitions, Polytopes, and Geometric Bijections
In 1857, Cayley showed that certain sequences, now called Cayley compositions, are equinumerous with certain partitions into powers of 2. In this paper we give a simple bijective proof of this result and a geometric generalization to equality of Ehrhart polynomials between two convex polytopes. We then apply our results to give a new proof of Braun’s conjecture proved recently by the authors [K...
متن کاملMinimum Convex Partitions and Maximum Empty Polytopes
Let S be a set of n points in Rd. A Steiner convex partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a Steiner convex partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first c...
متن کاملCayley Compositions, Partitions, Polytopes, and Geometric Bijections
In 1857, Cayley showed that certain sequences, now called Cayley compositions, are equinumerous with certain partitions into powers of 2. In this paper we give a simple bijective proof of this result and obtain several extensions. We then extend this bijection to an affine linear map between convex polyhedra to give and new proof of Braun’s conjecture. As an application, we give a slight improv...
متن کاملGradient superconvergence on uniform simplicial partitions of polytopes
Superconvergence of the gradient for the linear simplicial finite-element method applied to elliptic equations is a well known feature in one, two, and three space dimensions. In this paper we show that, in fact, there exists an elegant proof of this feature independent of the space dimension. As a result, superconvergence for dimensions four and up is proved simultaneously. The key ingredient ...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2019
ISSN: 1077-8926
DOI: 10.37236/8114