An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

Breuer, Felix

doi:10.1007/978-3-319-15081-9_1

Cited by 6 publications

(5 citation statements)

References 57 publications

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“…Polyhedral models have proven to be a useful tool in combinatorics [9,7,5] and in partition theory [4,11]. In particular, they can help in the construction of bijective proofs for partition identities, as the present article demonstrates, and even in the construction of combinatorial witnesses for partition congruences [10].…”

Section: Discussionmentioning

confidence: 76%

A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

Breuer¹,

Kronholm²

2016

Res. number theory

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The main result of this paper is a bijective proof showing that the generating function for partitions with bounded differences between largest and smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through analytic methods by Andrews, Beck, and Robbins. Our approach is geometric: We model partitions with bounded differences as lattice points in an infinite union of polyhedral cones. Surprisingly, this infinite union tiles a single simplicial cone. This construction then leads to a bijection that can be interpreted on a purely combinatorial level.

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Section: Discussionmentioning

confidence: 76%

A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

Breuer¹,

Kronholm²

2016

Res. number theory

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“…We first give a brief review of Ehrhart theory; see [3] for a more detailed introduction. Recall that for a set ∆ ⊆ R n , and a positive real number q, the q-dilation of ∆ is the set…”

Section: Proof Of the Existence Of The A-polynomialmentioning

confidence: 99%

Tutte polynomials for oriented matroids

Awan¹,

Bernardi²

2022

Preprint

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The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid N , we associate a polynomial invariant AN (q, y, z), which we call the A-polynomial. The A-polynomial has the following interesting properties among many others:• a specialization of AN gives the Tutte polynomial of the underlying unoriented matroid N ,• when the oriented matroid N corresponds to an unoriented matroid (that is, when the elements of the ground set come in pairs with opposite orientations), the invariant AN is equivalent to the Tutte polynomial of this unoriented matroid (up to a change of variables), • the invariant AN detects, among other things, whether N is acyclic and whether N is totally cyclic. We explore various properties and specializations of the A-polynomial. We show that some of the known properties or the Tutte polynomial of matroids can be extended to the A-polynomial of regular oriented matroids. For instance, we show that a specialization of AN counts all the acyclic orientations obtained by reorienting some elements of N , according to the number of reoriented elements.Let us mention that in a previous article we had defined an invariant of oriented graphs that we called the B-polynomial, which is also a generalization of the Tutte polynomial. However, the B-polynomial of an oriented graph N is not equivalent to its A-polynomial, and the B-polynomial cannot be extended to an invariant of regular oriented matroids.

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“…where the sum is over all lattice points which lie in some cone of C(F ). Since E C(F ) = F ∈F M type(C(F )) , this is a quasisymmetric function, first appearing in the work of Breuer (2015).…”

Section: Ehrhart Quasisymmetric Functionmentioning

confidence: 99%

Quasisymmetric Functions from Combinatorial Hopf Monoids and Ehrhart Theory

White¹

2016

Preprint

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We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.Resumé. Nous étudions les fonctions quasisymétriques associées aux monoïdes de Hopf combinatoriaux. Nous démontrons que ces invariants sont des objets naturels à la théorie de Ehrhart. De plus, certains correspondent à des fonctions de Hilbert associées à des complexes simpliciaux relatifs. Cette classe de complexes, constitue un mono de de Hopf, révélant ainsi un lien entre les algèbres de Hopf, la théorie de Ehrhart, et l'algèbre commutative. Nous étudions également diverses catégories de fonctions quasisymétriques.

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An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

Cited by 6 publications

References 57 publications

A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

Tutte polynomials for oriented matroids

Quasisymmetric Functions from Combinatorial Hopf Monoids and Ehrhart Theory

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