Asymptotic bounds on graphical partitions and partition comparability

  title={Asymptotic bounds on graphical partitions and partition comparability},
  author={Stephen Melczer and M. Michelen and S. Mukherjee},
  journal={arXiv: Combinatorics},
  • Stephen Melczer, M. Michelen, S. Mukherjee
  • Published 2020
  • Mathematics
  • arXiv: Combinatorics
  • 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


    A Recurrence for Counting Graphical Partitions
    • 19
    • PDF
    Confirming Two Conjectures About the Integer Partitions
    • B. Pittel
    • Computer Science, Mathematics
    • J. Comb. Theory, Ser. A
    • 1999
    • 38
    • PDF
    On graphical partitions
    • 27
    Dominance Order and Graphical Partitions
    • A. Kohnert
    • Mathematics, Computer Science
    • Electron. J. Comb.
    • 2004
    • 10
    • PDF
    A Note on Graphical Partitions
    • 9