Asymptotic joint distribution of the extremities of a random Young diagram and enumeration of graphical partitions

Abstract: An integer partition of $n$ is a decreasing sequence of positive integers that add up to $[n]$. Back in $1979$ Macdonald posed a question about the limit value of the probability that two partitions chosen uniformly at random, and independently of each other, are comparable in terms of the dominance order. In $1982$ Wilf conjectured that the uniformly random partition is a size-ordered degree sequence of a simple graph with the limit probability $0$. In $1997$ we showed that in both, seemingly unrelated, cases the limit probabilities are indeed zero, but our method left open the problem of convergence rates. The main result in this paper is that each of the probabilities is $e^{-0.11\log n/\log\log n}$, at most. A key element of the argument is a local limit theorem, with convergence rate, for the joint distribution of the $[n^{1/4-\varepsilon}]$ tallest columns and the $[n^{1/4-\varepsilon}]$ longest rows of the Young diagram representing the random partition.