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An integer partition of $n$ is a decreasing sequence of positive integersthat add up to $[n]$. Back in $1979$ Macdonald posed a question about the limitvalue of the probability that two partitions chosen uniformly at random, andindependently of each other, are comparable in terms of the dominance order. In$1982$ Wilf conjectured that the uniformly random partition is a size-ordereddegree sequence of a simple graph with the limit probability $0$. In $1997$ weshowed that in both, seemingly unrelated, cases the limit probabilities areindeed zero, but our method left open the problem of convergence rates. Themain result in this paper is that each of the probabilities is $e^{-0.11\logn/\log\log n}$, at most. A key element of the argument is a local limittheorem, with convergence rate, for the joint distribution of the$[n^{1/4-\varepsilon}]$ tallest columns and the $[n^{1/4-\varepsilon}]$ longestrows of the Young diagram representing the random partition.

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