How to Kill Time

Simon has already tackled all the problems from the contests and now spends his time practicing by arranging $n$ pebbles found near his computer into piles of varying sizes. Initially, he places all the pebbles one by one. Each subsequent minute, he rearranges them into the next lexicographic partition. The order of these partitions is determined as follows: consider all partitions of $n$ pebbles into piles, and in each partition, arrange the piles in non-decreasing order. We consider the partition a_1 + a_2 + \dots + a_n to be lexicographically smaller than b_1 + b_2 + \dots + b_m if, for some $k$, all j \leq k satisfy a_j = b_j, and either k = n, or a_{k+1} < b_{k+1}.