An integer partition is called graphical if it is the degree sequence of a simple graph. In this talk, we will show that the probability that a uniformly chosen partition of size $n$ is graphical, decreases to 0 faster than $n^{-0.003}$, thereby addressing a conjecture of Boris Pittel. A lower bound of $n^{-0.5}$ for this rate was already proved by Paul Erdős and Bruce Richmond in 1993, and so this demonstrates that the probability actually decreases polynomially. Key to our argument is an asymptotic characterization of the joint distribution of the leading rows and columns of the Young diagram of a uniformly random partition, combined with a celebrated characterization of graphical partitions due to Erdős and Gallai. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.

Zoom link: https://us02web.zoom.us/j/88670406480 Meeting ID: 886 7040 6480