Papers and Preprints (arxiv profile)
3. Bijectivity of a generalized Pak-Stanley labeling (with Olivier Bernardi), arXiv:2603.24886, Accepted to FPSAC 2026
Abstract: The Pak-Stanley labeling is a bijection between the regions of the $m$-Shi arrangement and the $m$-parking functions. Mazin generalized this labeling to every deformation of the braid arrangement and proved that this labeling is always surjective onto a set of directed multigraph parking functions. We provide a right inverse to the generalized Pak-Stanley labeling, and identify a class $\mathcal{C}$ of arrangements for which this labeling is bijective. The class $\mathcal{C}$ includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in $\mathcal{C}$ are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective.
2. The geometry of a counting formula for deformations of the braid arrangement (with Aaron Z. Lin), arXiv:2603.24885 (2026), submitted
Abstract: We consider real hyperplane arrangements whose hyperplanes are of the form $\{x_i - x_j = s\}$ for some integer $s$, which we call deformations of the braid arrangement. In 2018, Bernardi gave a counting formula for the number of regions of any deformation of the braid arrangement $\mathcal{A}$ as a signed sum over some decorated trees. He further showed that each of these decorated trees can be associated to a region $R$ of the arrangement $\mathcal{A}$, and hence we can consider the contribution of each region to the signed sum. Bernardi also implicitly showed that for transitive arrangements, the contribution of any region of the arrangement is $1$.
We remove the transitivity condition, showing that for any deformation of the braid arrangement the contribution of a region to the signed sum is $1$.
This provides an alternative proof of the original counting formula, and sheds light on the geometry underlying the formula. We further use this new geometric understanding to better understand the contribution of a tree.
Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements, arXiv:2506.00941, submitted. Extended abstract accepted to FPSAC 2025
arXiv:2506.00941, FPSAC Extended Abstract, Poster
Abstract: A recent result of Lofano and Paolini expresses the characteristic polynomial of a real hyperplane arrangement in terms of a projection statistic on the regions of the arrangement. We use this result to give an alternative proof for Greene and Zaslavsky's interpretation for the coefficients of the chromatic polynomial of a graph and further generalize this interpretation to signed graphs. We also show that this projection statistic has a nice combinatorial interpretation in the case of the braid arrangement, which generalizes to graphical arrangements of natural unit interval graphs.
Projects
The homomorphic chromatic number of signed graphs and co-chromatic number of graphs, MSc Project done under the guidance of N. Narayanan and Reza Naserasr