A geometric proof of the Brenti--Welker identity
We construct a hypersimplicial subdivision of the $r$-dilation of the $i$-th hypersimplex of dimension $d$ that provides a geometric proof of the Brenti--Welker identity.
math.CO↗
arXiv subjects
Ognjen Papaz
Publications and source records attributed to Ognjen Papaz.
Shelling of links and star clusters in edgewise subdivision of a simplex
We show that the combinatorial types of the links of the vertices in the edgewise triangulation $T_{k,q}$ of a $(k-1)$-simplex are encoded by the partitions of $k$. Each of these complexes is isomorphic to a subcomplex of the barycentric subdivision of the boundary of a $(k-1)$-simplex, and the containment relations among them are described by a new poset on the set of partitions of $k$. We compute the $h$-vectors of these complexes and determine the number of vertices of $T_{k,q}$ whose links are the same (correspond to the same partition). The combinatorial type of the link of an $(s-1)$-dimensional face of $T_{k,q}$ corresponds to a partition $(λ_1,λ_2,\ldots,λ_s)$ of $k$ into $s$ parts, together with additional partitions of each $λ_i$. We also enumerate the combinatorial types of all $m$-dimensional complexes that arise as the links in edgewise triangulations. A new permutation statistic, \textit{the faithful initial part}, is introduced and used to describe the star cluster of a facet of $T_{k,q}$. By examining a specific shelling of this star cluster, we prove that the $i$-th entry of its $h$-vector counts the number of permutations of $[k]$ with exactly $i$ descents, taking into account the faithful initial part as the multiplicity. Finally, we describe a concrete shelling order for $T_{k,q}$, give a combinatorial interpretation of its $h$-vector, and derive an explicit formula for it.
math.CO↗
A labeling of the Simplex-Lattice Hypergraph with at most 2 colors on each hyperedge
This paper provides a positive answer to the question of Mirzakhani and Vondrak that asks if there is a Sperner-admissible labeling of the simplex-lattice hypergraph such that each hyperedge uses at most 2 colors.
math.CO↗
Sperner's colorings of hypergraphs arising from edgewise triangulations
We investigate Sperner's labelings of $H^π_{k,q}$, the hypergraph whose hyperedges are facets of the edgewise triangulation of a $(k-1)$-simplex defined by a permutation $π\in \mathbb{S}_{k-1}$. Mirzakhani and Vondr\' ak showed that the greedy coloring of $H^{\mathrm{Id}}_{k,q}$ produces the maximal number of monochromatic hyperedges. The line graph of $H_{k,q}^π$ is built from the copies of the graph $G_π$ that represents which subsets of consecutive numbers of $[k-1]$ are contiguous in $π$. We characterize these graphs in terms of dissections a regular $k$-gon and also show how they encode the adjacency relation between a hypersimplex and the facets of its alcoved triangulation. The natural action of the dihedral group $D_k$ on a regular $k$-gon and graphs $G_π$ extends on the group of permutations $\mathbb S_{k-1}$. Independent sets of the graphs $G_π$ of the permutations that are not invariant under the rotation are used to define a class of Sperner's colorings that produce more monochromatic hyperedges then the greedy colorings. This colorings are also optimal for a certain permutations.
math.CO↗
On a generalization of compact and connected spaces
In this paper we will give two different natural generalizations of compact spaces and connected spaces simultaneously. We will show that these generalizations coincide for the subspaces of the real line and that they differ for subspaces of plane.
math.GN↗