Tabular prediction traditionally relies on gradient-boosted decision trees and deep learning models, which excel in specific tasks but lack interpretability and transferability. Reasoning large language models (LLMs) promise cross-task adaptability with transparent reasoning traces, yet their potential for tabular data remains unrealized. To bridge this gap, we propose a reasoning framework centered on Permutation Relative Policy Optimization (PRPO), a reinforcement learning method that encodes column-permutation invariance as a structural prior. By estimating advantages across label-preserving permutations, PRPO transforms sparse rewards into dense signals, activating latent numerical reasoning capabilities of LLMs with limited supervision. Extensive experiments show that our method matches fully supervised baselines and dominates in zero-shot settings, performing on par with 32-shot strong baselines. Remarkably, our 8B model significantly outperforms much larger LLMs, achieving up to a 53.17% improvement over DeepSeek-R1 (685B).