Combinatorial Reciprocity Theorems

Beck, Matthias

doi:10.1090/gsm/195

Cited by 31 publications

(17 citation statements)

References 29 publications

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“…Our next goal is to determine their Ehrhart quasi-polynomials for a particular interesting case. We briefly recall the basics of Ehrhart theory; for more see, for example, [2,3]. If P ⊂ R n is a d-dimensional polytope with rational vertex coordinates, then the function Ehr P (k) := |kP ∩ Z n | agrees with a quasi-polynomial of degree d. We will identify Ehr P (k) with this quasi-polynomial, called the Ehrhart quasi-polynomial.…”

Section: Canonical Subdivisionsmentioning

confidence: 99%

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Two Double Poset Polytopes

Chappell¹,

Friedl²,

Sanyal³

2017

SIAM J. Discrete Math.

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Abstract. To every poset P , Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P . This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (2011) introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the double order polytope and the double chain polytope and we amply demonstrate that they geometrically capture double posets, i.e., the interaction between the two partial orders. We describe the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets. We also describe a curious connection to Geissinger's valuation polytopes and we characterize 2-level polytopes among our double poset polytopes.Fulkerson's anti-blocking polytopes from combinatorial optimization subsume stable set polytopes of graphs and chain polytopes of posets. We determine the geometry of Minkowskiand Cayley sums of anti-blocking polytopes. In particular, we describe a canonical subdivision of Minkowski sums of anti-blocking polytopes that facilitates the computation of Ehrhart (quasi-)polynomials and volumes. This also yields canonical triangulations of double poset polytopes.Finally, we investigate the affine semigroup rings associated to double poset polytopes. We show that they have quadratic Gröbner bases, which gives an algebraic description of the unimodular flag triangulations described in the first part.

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Section: Canonical Subdivisionsmentioning

confidence: 99%

“…Associating ∆(J (P )) to a poset P is very natural and can be motivated, for example, from an algebraic-combinatorial approach to the order polynomial (cf. [3]). It would be very interesting to know if the association P to ∆ ni (P) is equally natural from a purely combinatorial perspective.…”

Section: Triangulations and Transfersmentioning

confidence: 99%

Two Double Poset Polytopes

Chappell¹,

Friedl²,

Sanyal³

2017

SIAM J. Discrete Math.

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“…Computing integer-point transforms of half-open simplicial cones is quite straightforward (and explicitly done in [1,Section 4.6]): For every 1 ≤ k ≤ m there is a bounded set P k ⊂ Z d × Z r , points v k,1 , . .…”

Section: I⊆[r]mentioning

confidence: 99%

“…Mixed volumes arise in virtually all mathematical disciplines and, most importantly, give rise to the deep theory of geometric inequalities; see, for example, Schneider [18]. Among the most fundamental properties, one trivially observes that MV d is symmetric and Minkowski additive in each argument and, not so trivially, that (1) 0 ≤ MV d (P 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Combinatorial mixed valuations

Jochemko

Sanyal

2017

Advances in Mathematics

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Combinatorial mixed valuations associated to translation-invariant valuations on polytopes are introduced. In contrast to the construction of mixed valuations via polarization, combinatorial mixed valuations reflect and often inherit properties of inhomogeneous valuations. In particular, it is shown that under mild assumptions combinatorial mixed valuations are monotone and hence nonnegative. For combinatorially positive valuations, this has strong computational implications. Applied to the discrete volume, the results generalize and strengthen work of Bihan (2015) on discrete mixed volumes. For rational polytopes, it is proved that combinatorial mixed monotonicity is equivalent to monotonicity. Stronger even, a conjecture is substantiated that combinatorial mixed monotonicity implies the homogeneous monotonicity in the sense of Bernig-Fu (2011).

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“…These two polynomials are related to each other by the following reciprocity theorem proved by Stanley ([10,11], see also [1,3,4] for a recent survey).…”

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confidence: 99%

Euler characteristic reciprocity for chromatic, flow and order polynomials

Miyatani¹,

Yoshinaga²

2017

jsing

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The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials.

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Combinatorial Reciprocity Theorems

Cited by 31 publications

References 29 publications

Two Double Poset Polytopes

Two Double Poset Polytopes

Combinatorial mixed valuations

Euler characteristic reciprocity for chromatic, flow and order polynomials

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