1,720,985 research outputs found
Representations of automorphism groups on the homology of matroids
Full text linkGiven a group G of automorphisms of a matroid M, we describe the representations of G on the homology of the independence complex of the dual matroid M∗ . These representations are related to the homology of the lattice of flats of M, and (when M is realizable) to the top cohomology of a hyperplane arrangement. Finally, we analyze in detail the case of the complete graph, which has applications to algebraic geometry
Graph colorings, flows and arithmetic Tutte polynomial
No full textAbstractWe 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]
On a conjecture of Hivert and Thiery about Steenrod operators
No full textWe prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics. In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture.We prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics (F. Hivert and N. Thiéry, 2004 [HT]). In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture. © 2012 Elsevier Inc
Combinatorics and topology of toric arrangements defined by root systems.
No full textGiven the toric (or toral) arrangement defined by a root system Φ, we describe the poset of its layers (connected components of intersections) and we count its elements. Indeed we show how to reduce to zero-dimensional layers, and in this case we provide an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of Φ. Then we compute the Euler characteristic and the Poincare' polynomial of the complement of the arrangement, which is the set of regular points of the torus
Arithmetic matroids, the Tutte polynomial and toric arrangements
Full text linkAbstractWe introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group. In particular, we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo’s formula for the classical Tutte polynomial
Graph colorings, flows and arithmetic Tutte polynomial
No full textWe 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.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]. © 2012 Elsevier Inc
Arithmetic matroids, the Tutte polynomial and toric arrangements
No full textWe introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group. In particular, we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial. © 2012 Elsevier Ltd
Ehrhart polynomial and arithmetic Tutte polynomial
No full textWe prove that the Ehrhart polynomial of a zonotope is a specialization of the arithmetic Tutte polynomial introduced by Moci (2012) [16]. We derive some formulae for the volume and the number of integer points of the zonotope. © 2012 Elsevier Ltd
On chromatic symmetric homology and planarity of graphs
Full text linkSazdanovic and Yip defined a categorification of Stanley's chromatic function
called the chromatic symmetric homology. In this paper we prove that (as
conjectured by Chandler, Sazdanovic, Stella and Yip), if a graph is
non-planar, then its chromatic symmetric homology in bidegree (1,0) contains
-torsion. Our proof follows a recursive argument based on
Kuratowsky's theorem.Comment: 11 pages. Some changes have been made in Section 3 in order to
improve readability. arXiv admin note: substantial text overlap with
arXiv:1911.13297 by other author
The expected jaggedness of order ideals
Full text linkThe jaggedness of an order ideal in a poset is the number of maximal elements in plus the number of minimal elements of not in . A probability distribution on the set of order ideals of is toggle-symmetric if for every , the probability that is maximal in equals the probability that is minimal not in . In this paper, we prove a formula for the expected jaggedness of an order ideal of under any toggle-symmetric probability distribution when is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan-López-Pflueger-Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp-Roby, of the antichain cardinality statistic for order ideals in partially ordered sets
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