Asymptotic Bounds on Graphical Partitions and Partition Comparability

Author:

Melczer Stephen1,Michelen Marcus2,Mukherjee Somabha3

Affiliation:

1. Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave W, Waterloo, Ontario N2L 3G1, Canada

2. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA

3. Department of Statistics, The Wharton School, University of Pennsylvania, 3730 Walnut Street, Philadelphia, PA 19104-3615, USA

Abstract

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.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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3. On the mathematical theory of isomers;Cayley;Philos. Mag.,1875

4. A simple proof of the Erd̋s–Gallai theorem on graph sequences;Choudum;Bull. Austral. Math. Soc.,1986

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