Asymptotic Bounds on Graphical Partitions and Partition Comparability
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Abstract An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by Erd̋s and Richmond, meaning our work demonstrates that the probability decreases polynomially. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.
2000 ◽
Vol 32
(01)
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pp. 244-255
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2000 ◽
Vol 32
(1)
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pp. 244-255
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1998 ◽
Vol 58
(1)
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pp. 1-13
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Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop
2016 ◽
Vol 26
(12)
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pp. 1650204
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1953 ◽
Vol 49
(1)
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pp. 59-62
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