Nicolas Avila | Searcharxiv

arXiv2026

Luck and magic for Pitman-Stanley polytopes and parking functions

Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart polynomials of these polytopes, which in turn implies that their $h^*$-polynomials are real-rooted (and thus log-concave and unimodal). Our result is achieved by interpreting the coefficients of these Ehrhart polynomials in the \emph{magic basis} in terms of the number of \emph{lucky cars} in a modified parking protocol. Furthermore, we address the magic positivity problem for $\mathbf{y}$-generalized permutohedra and also discuss a \emph{magic} combinatorial interpretation for them, under the assumption that the input parameters are sufficiently large.

math.CO↗

arXiv2022

Energy Pricing in P2P Energy Systems Using Reinforcement Learning

The increase in renewable energy on the consumer side gives place to new dynamics in the energy grids. Participants in a microgrid can produce energy and trade it with their peers (peer-to-peer) with the permission of the energy provider. In such a scenario, the stochastic nature of distributed renewable energy generators and energy consumption increases the complexity of defining fair prices for buying and selling energy. In this study, we introduce a reinforcement learning framework to help solve this issue by training an agent to set the prices that maximize the profit of all components in the microgrid, aiming to facilitate the implementation of P2P grids in real-life scenarios. The microgrid considers consumers, prosumers, the service provider, and a community battery. Experimental results on the \textit{Pymgrid} dataset show a successful approach to price optimization for all components in the microgrid. The proposed framework ensures flexibility to account for the interest of these components, as well as the ratio of consumers and prosumers in the microgrid. The results also examine the effect of changing the capacity of the community battery on the profit of the system. The implementation code is available \href{https://github.com/Artifitialleap-MBZUAI/rl-p2p-price-prediction}{here}.

cs.LG↗