arXiv subjects

Vladimir Grujić

Publications and source records attributed to Vladimir Grujić.

9 recordsLinked to original sources

arXiv2026

Characteristic numbers of canonical toric manifolds and their applications

We compute all the Chern, Milnor and Pontryagin numbers for canonical toric manifolds associated with abstract simplicial complexes and the Stiefel-Whitney numbers for their real counterparts. Applications include combinatorial characterizations of the unitary, oriented and unoriented bordism classes, new geometrical representatives of the unitary bordism ring generators, a combinatorial criterion for a canonical toric manifold to bound, as well as the dimension estimates for their immersions into euclidean spaces.

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arXiv2024

The Redei-Berge Hopf algebra of digraphs

In a series of recent talks Richard Stanley introduced a symmetric function associated to digraphs called the Redei-Berge symmetric function. This symmetric function enumerates descent sets of permutations corresponding to digraphs. We show that such constructed symmetric function arises from a suitable structure of combinatorial Hopf algebra on digraphs. The induced Redei-Berge polynomial satisfies the deletion-contraction property which makes it similar to the chromatic polynomial. The Berge's classical result on the number of Hamiltonian paths in digraphs is a consequence of the reciprocity formula for the Redei-Berge polynomial.

math.CO↗

arXiv2021

Fundamental groups of 3-dimensional small covers

Small covers arising from 3-dimensional simple polytopes are an interesting class of 3-manifolds. The fundamental group is a rigid invariant for wide classes of 3-manifolds, particularly for orientable Haken manifolds, which include orientable small covers. By using Morse-theoretic approach we give a procedure to get an explicit, balanced presentation of the fundamental group of a closed, orientable 3-dimensional simple cover with minimal number of generators. Beside that the minimal Heegaard splitting is determined by this presentation.

math.GT↗

arXiv2019

Weighted $\mathsf{P}-$partitions enumerator

To an extended permutohedron we associate the weighted integer points enumerator, whose principal specialization is the $f$-polynomial. In the case of poset cones it refines Gessel's $\mathsf{P}$-partitions enumerator. We show that this enumerator is a quasisymmetric function obtained by universal morphism from the Hopf algebra of posets.

math.CO↗

arXiv2017

Weighted quasisymmetric enumerator for generalized permutohedra

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

math.CO↗

arXiv2017

Counting faces of nestohedra

A new algebraic formula for the numbers of faces of nestohedra is obtained. The enumerator function $F(P_B)$ of positive lattice points in interiors of maximal cones of the normal fan of the nestohedron $P_B$ 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 $F_q(P_B)$ and derive its determining recurrence relations. The $f$-polynomial of the nestohedron $P_B$ appears as the principal specialization of the quasisymmetric function $F_q(P_B)$.

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arXiv2017

Quasisymmetric functions for nestohedra

For a generalized permutohedron $Q$ the enumerator $F(Q)$ of positive lattice points in interiors of maximal cones of the normal fan $Σ_Q$ 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.

math.CO↗

arXiv2017

Counting faces of graphical zonotopes

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

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arXiv2012

Hopf algebra of building sets

The combinatorial Hopf algebra on building sets $BSet$ 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 $BSet$, 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 $\mathbf{c}\mathbf{d}-$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.

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