Counting Cycles with AI

Despite recent progress, AI still struggles with advanced mathematics. We consider a difficult open problem: How to derive a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistic? The CEEF problem has no known general solution, and requires delicate combinatorics and tedious calculations. Such a task is hard to accomplish by humans but is an ideal example where AI can be very helpful. We solve the problem by combining our novel approach with the powerful coding skills of AI. Our results use delicate graph theory and contain new formulas for general cases that were not previously known. We find that, while AI is unable to solve the problem on its own, it is able to solve it if we provide it with a clear strategy, step-by-step guidance, and carefully written prompts. For simplicity, we focus our study on DeepSeek-R1 but we also investigate other representative LLMs.