Asymptotic Bounds on Graphical Partitions and Partition Comparability

An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size n is graphical decreases to zero faster than n(-.003), answering a question of Pittel. A lower bound of n(-1/2) was proven by Erdos and Richmond, meaning our work demonstrates that the probability decreases polynomially. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.