Ehrhart polynomial and arithmetic Tutte polynomial

D'Adderio, Michele; Moci, Luca

doi:10.1016/j.ejc.2012.02.006

Cited by 32 publications

(17 citation statements)

References 23 publications

(30 reference statements)

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“…is in the ideal of R generated by the elements u ν,1 + u ν,2 and v ν,1 + v ν,2 for ν ∈ Λ. Then the above relation is a consequence of relations (17), (18) and (19).…”

Section: Comparison Tomentioning

confidence: 93%

“…(3) If M e,f has U 1,2 as its underlying matroid and M e,f does not then a dual argument applies. In all three cases, the relation given by Proposition 4.4 is a consequence of relations (17), (18) and (19), and we are done.…”

Section: Comparison Tomentioning

confidence: 99%

“…If R is a domain then this means that these two quantities are non-zero. If this condition is satisfied then equations (17), (18) and (19) are equivalent to (20) u…”

Section: Comparison Tomentioning

confidence: 99%

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Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

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This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids. We also produce some new invariants along with new convolution formulae.obtain several convolution formulae for different classes of combinatorial objects. Some of these formulae are well known, while others are (to the best of our knowledge) new: see for instance Propositions 5.12, 5.13 and 5.14 for the classical Tutte polynomial, Proposition 6.17 for the Las Vergnas polynomial, Proposition 6.20 for the Bollobás-Riordan polynomial and Theorem 10.9 for the arithmetic Tutte polynomial.We have chosen to exemplify our results with mostly matroid-like combinatorial structures, which are arguably simpler to deal with than topological examples. Our formalism, and in particular the mechanism of twist maps, should help uncover new invariants in the latter class and produce interesting convolution formulae. This will be the subject of a subsequent article.Layout. The structure of this paper is as follows. In Section 2 we give the definitions of minors systems and comonoids in set species. Section 3 contains the main results, including the definitions and statements of our universal invariants and formulae. Section 4 presents a number of further results which are less essential for our main developments.The remainder of the paper, Sections 5 through 10, comprises a sequence of applications of our theory to numerous individual minors systems. We work out Grothendieck groups, universal Tutte characters, and in many cases universal convolution formulae, and point out how these specialize to invariants and formulae present in the literature. Section 5 covers the minors system of matroids, of which all the subsequent sections are in one way or another generalizations, together with the minors system of graphs. Of the following sections, Section 6 is on delta-matroids, matroid perspectives, and their ilk is called on in Section 7 on relative matroids, but there are no (or at most incidental) dependences between these and Section 8 on polymatroids and generalised permutohedra, Section 9 on colored matroids, or Section 10 on arithmetic matroids, nor among the latter three, so the reader should have no trouble taking these in any order. The last four sections are also new by comparison with [37].Notation. We fix a commutative ring with unit K...

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“…is in the ideal of R generated by the elements u ν,1 + u ν,2 and v ν,1 + v ν,2 for ν ∈ Λ. Then the above relation is a consequence of relations (17), (18) and (19).…”

Section: Comparison Tomentioning

confidence: 93%

Section: Comparison Tomentioning

confidence: 99%

See 1 more Smart Citation

Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

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“…, (k) . In this case the matroid M G, is the direct sum of the matroids M G (1) , (1) , M G (2) , (2) , . .…”

Section: Proof Of Theorem 31mentioning

confidence: 99%

“…In [5] a new polynomial has been introduced, which provides a natural counterpart for toric arrangements of the Tutte polynomial of a hyperplane arrangement. In fact in [5] and [1] it has been shown how this polynomial has several applications to toric arrangements, vector partition functions and zonotopes.…”

Section: Introductionmentioning

confidence: 99%

Graph colorings, flows and arithmetic Tutte polynomial

D'Adderio

Moci

2013

Journal of Combinatorial Theory, Series A

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We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte (1954) [9].It is well known how to associate a matroid, and hence a Tutte polynomial, to a graph. Moreover, several combinatorial objects associated to a graph are counted by suitable specializations of this important invariant: (proper) q-colorings and (nowhere zero) q-flows are two of the most classical examples. See [6] and [10] for systematic accounts, or [1] for more recent developments.In [7] a new polynomial has been introduced, which provides a natural counterpart for toric arrangements of the Tutte polynomial of a hyperplane arrangement. In fact in [7] and [3] it has been shown how this polynomial has several applications to toric arrangements, vector partition functions and zonotopes.In [4] we introduced the notion of an arithmetic matroid, which generalizes that of a matroid, and whose main example is provided by a list of elements in a finitely generated abelian group. To this object we associated an arithmetic Tutte polynomial (which is the polynomial in [7] for the main example) and provided a combinatorial interpretation of it which extends the one given by Crapo for the classical Tutte polynomial.Encouraged by all this evidence (cf. also [5] where two parallel theories are developed), we think of the arithmetic matroids and the arithmetic Tutte polynomial as natural generalizations of their classical counterparts. So it seemed also natural to us to look for applications in graph theory.

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