arXiv subjects

Warut Thawinrak

Publications and source records attributed to Warut Thawinrak.

5 recordsLinked to original sources

arXiv2026

Counting Lattice Points in Minkowski Sums of Cross Polytopes

Motivated by Postnikov's study of lattice-point enumeration in Minkowski sums of simplices, we investigate lattice points in Minkowski sums of cross polytopes and establish analogous results, together with several related consequences. In particular, we introduce the support-enumerator associated with Postnikov's notion of draconian sequences and show that it coincides with the $h^*$-polynomial of the corresponding root polytope. This provides a new interpretation of the $h^*$-polynomial and yields a simple method for computing the volume of the corresponding polytope. By exploiting the symmetry of these root polytopes, we further establish a duality property for support-enumerators, which in turn provides a proof of a conjecture by Athanasiadis and Chapoton concerning the $h$-polynomials of preorders. Consequently, we obtain a formula for the number of lattice points in Minkowski sums of cross polytopes in terms of draconian sequences and show that these polytopes are Ehrhart positive. Furthermore, this formula leads to analogous expressions for the number of lattice points on their boundaries and for their surface volumes.

math.CO↗

arXiv2026

Counting Lattice Points in Generalized Permutohedra From A to B

We derive a formula for the number of lattice points in type B generalized permutohedra, providing a concise alternative to the formula obtained recently by Eur, Fink, Larson, and Spink as a result from a study of delta-matroids. Our approach builds upon the existing framework and techniques introduced by Postnikov in his work on type A generalized permutohedra, a family of polytopes interconnected with many mathematical concepts such as matroids and Weyl groups. In particular, we express the number of lattice points in type B generalized permutohedra in terms of Postnikov's notion of G-draconian sequences, from which their Ehrhart polynomials and volume formula follow as consequences.

math.CO↗

arXiv2025

Parking Function Polytopes

We extend the notion of parking function polytopes and study their geometric and combinatorial structure, including normal fans, face posets, and $h$-polynomials, as well as their connections to other classes of polytopes. To capture their combinatorial features, we introduce generalizations of ordered set partitions, called binary partitions and skewed binary partitions. Using properties of preorder cones, we characterize the skewed binary partitions that are in bijection with the cones of the normal fan of a parking function polytope. This description of the normal fan yields an explicit formula for the $h$-polynomials of simple parking function polytopes in terms of generalized Eulerian polynomials. Finally, we relate parking function polytopes to several well-known polytopes, leading to additional results, including formulas for their volumes and Ehrhart polynomials.

math.CO↗

arXiv2024

A Short Proof for the Polynomiality of the Stretched Littlewood-Richardson Coefficients

The stretched Littlewood-Richardson coefficient $c^{tν}_{tλ,tμ}$ was conjectured by King, Tollu, and Toumazet to be a polynomial function in $t.$ It was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg's formula, the hive conditions, and the Kostant partition function to show a stronger result that $c^ν_{λ,μ}$ is indeed a polynomial in variables $ν, λ, μ$ provided they lie in certain polyhedral cones. Motivated by Rassart's approach, we give a short alternative proof of the polynomiality of $c^{tν}_{tλ,tμ}$ using Steinberg's formula and a simple argument about the chamber complex of the Kostant partition function.

math.CO↗

arXiv2017

High-order filtered schemes for the Hamilton-Jacobi continuum limit of nondominated sorting

We investigate high-order finite difference schemes for the Hamilton-Jacobi equation continuum limit of nondominated sorting. Nondominated sorting is an algorithm for sorting points in Euclidean space into layers by repeatedly removing minimal elements. It is widely used in multi-objective optimization, which finds applications in many scientific and engineering contexts, including machine learning. In this paper, we show how to construct filtered schemes, which combine high order possibly unstable schemes with first order monotone schemes in a way that guarantees stability and convergence while enjoying the additional accuracy of the higher order scheme in regions where the solution is smooth. We prove that our filtered schemes are stable and converge to the viscosity solution of the Hamilton-Jacobi equation, and we provide numerical simulations to investigate the rate of convergence of the new schemes.

math.NA↗