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Confirming two conjectures about the integer partitions. (English) Zbl 0937.05008
The author answers a question posed by I. G. Macdonald in [Symmetric functions and Hall polynomials. 2nd ed. (Clarendon Press, Oxford) (1995; Zbl 0824.05059)]. He proves that if two partitions, $$\lambda$$ and $$\mu$$, are chosen uniformly at random and independent of each other from the set of partitions of $$n$$, then the probability that $$\lambda$$ and $$\mu$$ are comparable in the usual dominance order approaches 0 as $$n$$ approaches infinity. From the Gale-Ryser theorem, it follows that if $$\pi_n$$ is the probability that there exists a bipartite graph on $$(X,Y)$$ such that $$\lambda$$ and $$\mu$$ are the degree sequences of the respective vertex sets, then $$\lim_{n\to \infty} \pi_n = 0$$. The same methods enable the author to prove a conjecture made by Wilf in 1982: The probability that a randomly chosen partition of $$n$$ is the degree sequence of a graph approaches 0 as $$n$$ approaches infinity.

##### MSC:
 05A17 Combinatorial aspects of partitions of integers 05C07 Vertex degrees 60F05 Central limit and other weak theorems 60F20 Zero-one laws 60C05 Combinatorial probability
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##### References:
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