arXiv subjects

Gjergji Zaimi

Publications and source records attributed to Gjergji Zaimi.

7 recordsLinked to original sources

arXiv2025

Extremal diameters of 3-coloring graphs of trees

Given a tree $T$, its 3-coloring graph $\mathcal{C}_3(T)$ has as vertices the proper 3-colorings of $T$, with edges joining colorings that differ at exactly one vertex. We call the diameter of $\mathcal{C}_3(T)$ the 3-coloring diameter of $T$. We introduce the notion of balanced labelings of $T$ and show that the 3-coloring diameter equals the maximum $L_1$-norm of a balanced labeling. Using this equivalence, we determine the maximum and minimum values of the 3-coloring diameter over all trees on $n$ vertices and characterize the extremal trees.

math.CO↗

arXiv2023

Restricted Birkhoff polytopes and Ehrhart period collapse

We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the "longest increasing subsequence" have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand-Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson-Schensted-Knuth correspondence.

math.CO↗

arXiv2023

Combinatorial reciprocity for non-intersecting paths

We prove a combinatorial reciprocity theorem for the enumeration of non-intersecting paths in a linearly growing sequence of acyclic planar networks. We explain two applications of this theorem: reciprocity for fans of bounded Dyck paths, and reciprocity for Schur function evaluations with repeated values.

math.CO↗

arXiv2019

On the classification of higher Auslander algebras for Nakayama algebras

We give new improved bounds for the dominant dimension of Nakayama algebras and use those bounds to give a classification of Nakayama algebras with $n$ simple modules that are higher Auslander algebras with global dimension at least $n$. The classification is then used to extend the results on the inequality for the global dimension of Nakayama algebras obtained in \cite{MM}.

math.RT↗

arXiv2016

Frankl's Conjecture for subgroup lattices

We show that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.

math.CO↗

arXiv2015

The Topology of Equivariant Hilbert Schemes

For $G$ a finite group acting linearly on $\mathbb{A}^2$, the equivariant Hilbert scheme $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ is a natural resolution of singularities of $\operatorname{Sym}^r(\mathbb{A}^2/G)$. In this paper we study the topology of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ for abelian $G$ and how it depends on the group $G$. We prove that the topological invariants of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ are periodic or quasipolynomial in the order of the group $G$ as $G$ varies over certain families of abelian subgroups of $GL_2$. This is done by using the Bialynicki-Birula decomposition to compute topological invariants in terms of the combinatorics of a certain set of partitions.

math.AG↗

arXiv2014

Integrality of relative BPS state counts of toric Del Pezzo surfaces

Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert and conjectured by the authors to be integers. For toric Del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng; and more generally for toric Calabi-Yau threefolds by Konishi.

math.AG↗