Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8904832 | Advances in Mathematics | 2018 | 27 Pages |
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 eâ0.11logâ¡n/logâ¡logâ¡n, at most. A key element of the argument is a local limit theorem, with convergence rate, for the joint distribution of the [n1/4âε] tallest columns and the [n1/4âε] longest rows of the Young diagram representing the random partition.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Boris Pittel,