# Weighted Ehrhart functions

## TL;DR

This paper introduces an algorithm for computing weighted Ehrhart functions of lattice polytopes with polynomial weights, utilizing Lagrange interpolation, generating functions, and integer programming, with applications to enumeration problems.

## Contribution

It presents a novel algorithm for weighted Ehrhart functions, combining interpolation, generating functions, and algebraic methods, advancing computational tools in lattice polytope enumeration.

## Key findings

- Algorithm efficiently computes weighted Ehrhart functions.
- Uses generating functions and Eulerian numbers for polynomial weights.
- Applies integer programming to analyze Ehrhart ring properties.

## Abstract

We give an algorithm to compute weighted Ehrhart functions of lattice polytopes for polynomial weights using Lagrange interpolation. We show how to compute generating functions of polynomials using those of unit cubes and Eulerian numbers, and use integer programming to study the algebraic properties of the Ehrhart ring of the $d$-th unit cube. Then we show some applications to weighted Ehrhart functions and enumeration problems.