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Papers by Vladimir Grujić

arXiv (Cornell University), Mar 26, 2017

A new algebraic formula for the numbers of faces of nestohedra is obtained. The enumerator functi... more A new algebraic formula for the numbers of faces of nestohedra is obtained. The enumerator function F (PB) of positive lattice points in interiors of maximal cones of the normal fan of the nestohedron PB associated to a building set B is described as a morphism from the certain combinatorial Hopf algebra of building sets to quasisymmetric functions. We define the q-analog Fq(PB) and derive its determining recurrence relations. The f-polynomial of the nestohedron PB appears as the principal specialization of the quasisymmetric function Fq(PB).

arXiv: Combinatorics, 2016

It is a classical fact that the number of vertices of the graphical zonotope ZGammaZ_\GammaZGamma is equal ... more It is a classical fact that the number of vertices of the graphical zonotope ZGammaZ_\GammaZGamma is equal to the number of acyclic orientations of a graph Gamma\GammaGamma. We show that the fff-polynomial of ZGammaZ_\GammaZGamma is obtained as the principal specialization of the qqq-analog of the chromatic symmetric function of Gamma\GammaGamma.

SIAM Journal on Discrete Mathematics, 2017

For a generalized permutohedron Q the enumerator F (Q) of positive lattice points in interiors of... more For a generalized permutohedron Q the enumerator F (Q) of positive lattice points in interiors of maximal cones of the normal fan ΣQ is a quasisymmetric function. We describe this function for the class of nestohedra as a Hopf algebra morphism from a combinatorial Hopf algebra of building sets. For the class of graph-associahedra the corresponding quasisymmetric function is a new isomorphism invariant of graphs. The obtained invariant is quite natural as it is the generating function of ordered colorings of graphs and satisfies the recurrence relation with respect to deletions of vertices.

Ars Mathematica Contemporanea, 2017

It is a classical fact that the number of vertices of the graphical zonotope ZΓ is equal to the n... more It is a classical fact that the number of vertices of the graphical zonotope ZΓ is equal to the number of acyclic orientations of a graph Γ. We show that the f-polynomial of ZΓ is obtained as the principal specialization of the q-analog of the chromatic symmetric function of Γ.

The Electronic Journal of Combinatorics, 2012

The combinatorial Hopf algebra on building sets BSetBSetBSet extends the chromatic Hopf algebra of simp... more The combinatorial Hopf algebra on building sets BSetBSetBSet extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized Dehn-Sommerville relations for the Hopf algebra BSetBSetBSet, we define a class of building sets called eulerian and show that eulerian building sets satisfy Bayer-Billera relations. We show the existence of the mathbfcmathbfd−\mathbf{c}\mathbf{d}-mathbfcmathbfd−index, the polynomial in two noncommutative variables associated to an eulerian building set. The complete characterization of eulerian building sets is given in terms of combinatorics of intersection posets of antichains of finite sets.

Journal of Algebraic Combinatorics, 2019

We introduce a weighted quasisymmetric enumerator function associated with generalized permutohed... more We introduce a weighted quasisymmetric enumerator function associated with generalized permutohedra. It refines the Billera, Jia and Reiner quasisymmetric function which also includes the Stanley chromatic symmetric function. Besides that, it carries information of face numbers of generalized permutohedra. We consider more systematically the cases of nestohedra and matroid base polytopes.

European Journal of Mathematics, 2016

The generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf... more The generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf algebra of hypergraphs. We characterize a wide class of eulerian hypergraphs according to the combinatorics of underlying clutters. The analogous results hold for simplicial complexes by the isomorphism which is induced from the correspondence of clutters and simplicial complexes.

We construct small covers and quasitoric manifolds over a given nnn-colored simple polytope PnP^nPn... more We construct small covers and quasitoric manifolds over a given nnn-colored simple polytope PnP^nPn with interesting properties. Their Stiefel-Whitney classes are calculated and used as obstruction to immersions and embeddings into Euclidean spaces. In the case nnn is a power of two we get the sharpest bounds.

Israel Journal of Mathematics, 2003

We study some of the combinatorial structures related to the signature of G-symmetric products of... more We study some of the combinatorial structures related to the signature of G-symmetric products of (open) surfaces SP m G (M) = M m /G where G ⊂ Sm. The attention is focused on the question what information about a surface M can be recovered from a symmetric product SP n (M). The problem is motivated in part by the study of locally Euclidean topological commutative (m + k, m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SP m G (M)) in terms of the cycle index Z(G;x) of G, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfaces M g,k , M g ′ ,k ′ such that the manifolds SP m (M g,k) and SP m (M g ′ ,k ′) are often not homeomorphic, although they always have the same homotopy type provided 2g + k = 2g ′ + k ′ and k, k ′ ≥ 1.

Publications de l'Institut Math?matique (Belgrade), 2003

We deal with Hirzebruch genera of complete intersections of non-singular projective hyper surface... more We deal with Hirzebruch genera of complete intersections of non-singular projective hyper surfaces. We give the formula for genera of algebraic curve and surfaces and prove that symmetric squares of algebraic curve of genus g > 0 are not projective complete intersections.

Topology and its Applications, 2005

We study the combinatorics and topology of general arrangements of subspaces of the form D + SP n... more We study the combinatorics and topology of general arrangements of subspaces of the form D + SP n−d (X) in symmetric products SP n (X) where D ∈ SP d (X). Symmetric products SP m (X) := X m /Sm, also known as the spaces of effective "divisors" of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [7] and [21] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [34] and [37], we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP n (M g,k), where M g,k is a Riemann surface of genus g punctured at k points, a problem which was originally motivated by the study of commutative (m + k, m)-groups [32].