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

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...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2018-05, Vol.330, p.280-306
Main Author: Pittel, Boris
Format: Article
Language:English
Subjects:
Convergence rate
Enumeration
Extremities
Graphical
Partition
Young diagram
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Summary: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−0.11log⁡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 [n1/4−ε] tallest columns and the [n1/4−ε] longest rows of the Young diagram representing the random partition.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2018.03.013