Mixed Volumes and Slices of the Cube

Ehrenborg, Richard; Readdy, Margaret; Steingrı́msson, Einar

doi:10.1006/jcta.1997.2832

Cited by 17 publications

(18 citation statements)

References 5 publications

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“…Note that by definition a j (d, k) = 0 for k < 0, j < 1, k > d − 1 and j > d. As far as we know, the (A, j)-Eulerian polynomials were first considered by Brenti and Welker [8], though the (A, j)-Eulerian numbers and generalizations of them were considered earlier (see, e.g., [9,26]). 1 A simplex is a d-polytope with (the minimal number of) d + 1 vertices; it is unimodular if these vertices have integer coordinates and the simplex has (minimal) volume 1 d!…”

Section: Descent Statisticsmentioning

confidence: 99%

$h^\ast $-polynomials of zonotopes

Beck

Jochemko

McCullough

2018

Trans. Amer. Math. Soc.

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The Ehrhart polynomial of a lattice polytope P encodes information about the number of integer lattice points in positive integral dilates of P . The h * -polynomial of P is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the h *polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the h * -polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the h * -polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley (1991) on the Ehrhart polynomial. Our results hold not only for h * -polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all h * -polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.

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Section: Descent Statisticsmentioning

confidence: 99%

$h^\ast $-polynomials of zonotopes

Beck

Jochemko

McCullough

2018

Trans. Amer. Math. Soc.

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“…The formula may be found by multiplying both sides of the well-known identity ∞ k=0 k r · t k = r −1 i=0 A r,i t i+1 (1 − t) r +1 (see, e.g., Section 6.5 of [6]) by (1 − t) −1 . This approach may also yield some interesting formulas for cubical posets, since the Eulerian numbers have an interpretation as volumes of certain slices of a hypercube (see [22] and the generalization by Ehrenborg et al [13]). Another interesting approach to Eulerian numbers may be found in the recent work of Lam and Postnikov [19].…”

Section: Discussionmentioning

confidence: 99%

The Stirling Polynomial of a Simplicial Complex

Hetyei

2005

Discrete Comput Geom

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We introduce a new encoding of the face numbers of a simplicial complex, its Stirling polynomial, that has a simple expression obtained by multiplying each face number with an appropriate generalized binomial coefficient. We prove that the face numbers of the barycentric subdivision of the free join of two CW -complexes may be found by multiplying the Stirling polynomials of the barycentric subdivisions of the original complexes. We show that the Stirling polynomial of the order complex of any simplicial poset and of certain graded planar posets has non-negative coefficients. By calculating the Stirling polynomial of the order complex of the r -cubical lattice of rank n + 1 in two ways, we provide a combinatorial proof for the following identity of Bernoulli polynomials:

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“…The refinement of the Eulerian number, say A d,k, j , is the number of permutations in the symmetric group S d with k descents and ending with the element j. To give the combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube, in [7], the authors find A d,k, j are associated to the volume of the slices of unit cubes, i.e.,…”

Section: B-splines and Refined Eulerian Numbersmentioning

confidence: 99%

“…Here, S n d stands for the set of all indexed permutations which is an ordinary permutation in the symmetric group S d where each letter has been assigned an integer between 0 and n − 1 (see [4,7]). In [4], Steingrímsson showed that…”

Section: B-splines and Refined Eulerian Numbersmentioning

confidence: 99%

Eulerian numbers: A spline perspective

Wang

2010

Journal of Mathematical Analysis and Applications

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Mixed Volumes and Slices of the Cube

Cited by 17 publications

References 5 publications

$h^\ast $-polynomials of zonotopes

$h^\ast $-polynomials of zonotopes

The Stirling Polynomial of a Simplicial Complex

Eulerian numbers: A spline perspective

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