On the average time complexity of computation with random partition

Some computations are based on structures of random partition. They take an n-size problem as input, then break this problem into sub-problems of randomized size, execute calculations on each sub-problems and combine results from these calculations at last. We propose a combinatorial method for analyzing such computations and prove that the averaged time complexity is in terms of Stirling numbers of the second kind. The result shows that the average time complexity is decreased about one order of magnitude compared to that of the original solution. We also show two application cases where random partition structures are applied to improve performance.