Asymptotic bounds on graphical partitions and partition comparability

@article{Melczer2020AsymptoticBO,
title={Asymptotic bounds on graphical partitions and partition comparability},
author={Stephen Melczer and M. Michelen and S. Mukherjee},
journal={arXiv: Combinatorics},
year={2020}
}

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 Erd\H{o}s and Richmond, and so this demonstrates that the probability decreases polynomially. Key to our argument is an asymptotic result of Pittel characterizing the joint distribution of the first rows… CONTINUE READING